数理解析研究所講究録 1345 巻 2003 年 127-147

127

Pregeometric Shellsof a Rational Quartic Curveand of a Veronese surface
姫工大

遊佐

理

毅 (Takeshi Usa)

Dept. of Math. Himeji Institute of Technology*

Abstract
This is a resume of the author’s talk given at R.I.M.S. on June 182003 and is a partial revision of a
recent paper [23] with additional results. The theme of that talk was to give a supporting evidence
for general conjectures raised in [20] (cf. Conjecture 0.1). We classify all the pregeometric shells ofa rational normal quartic curve $X$ (resp. of a Veronese surface $X$ ), namely the closed subschemes(resp. inin) which include $X$ and whose homogeneous coordinate rings satisfy the
injectivity condition. This
injectivity condition is the same as to impose that for every nonnegative integer , minimal generators of -th syzygy of its homogeneous coordinate ring form a partof minimal generators of -th syzygy of the homogeneous coordinate ringof $X$ . We see that
all those pregeo metric shells turn out to be reduced and irreducible, and moreover the varieties ofA-genus zero (embedded by their complete linear systems, cf. Remark 1.3), namely the varieties ofminimal degree, as predicted from these general conjectures.
Keywords: pregeometric shell, rational quartic curve, Veronese surface, -genus, variety of minimal
degree$\mathrm{P}^{5}(\mathbb{C})$$\mathrm{P}^{4}(\mathbb{C})$$\mathrm{T}\mathrm{o}\mathrm{r}$$\mathrm{T}\mathrm{o}\mathrm{r}$

$q$

$q$

$R_{\mathit{1}\iota}$$q$$\triangle$\S 0 Introduction.
Dreaming to construct a theory of projective embeddings modeled after the classical Galois theory, we
presented several conjectures and problems on the geo metric structures of projective embeddings in ourprevious paper [20]. One of the key concepts appeared in these conjectures and problems is “pregeometric-shell; cf. Definition 2.4), which was first introduced in [19] with expectation that itshell” (abbr.
may play a similar role as a concept of “intermediate extension field” in the Galois theory and rnay cut
our way to the dream.
We already saw in [20] and [22] that the (pre)geometric shells inherit many excellent properties (cf
Proposition 2.5, Corollary 3.7) from their (pre)geometric core (abbr.
core ; cf. Definition 2.4),
which reflect the structure of higher syzygies of the homogeneous coordinate ring of the (pre)geometriccore. “Pregeometric shells” had appeared implicitly in many classical works (cf. [13], [15], [5], [7], [16],$[6],[3]$ etc.) as actual examples. However, there are still deep mysteries on pregeometric shells left (cf.
[20] . For example, the following conjectures are still open with slight modifications (precisely, only the
first one is added after publishing of [20] .
$\mathrm{P}\mathrm{G}$$(\mathrm{P})\mathrm{G}$

$)$

$)$

’2167 Shosha, Himeji, 671-2201 Japan.
-mail address : usa@sci.himeji-tech.ac.jp
Typeset by-BTEX$\mathrm{E}$$A\mathrm{A}4\mathrm{S}$128

Conjecture 0.1 We fix the total spacewith the tautological ample line bundle $O_{P}(1)=$$O_{P}(H)$ and consider its closed subschemes defined overwith induced polarizations.
$P=\mathrm{P}^{N}(\mathbb{C})$$\mathbb{C}$(0.1.1)

Assume that a closed subscheme $W$ is a pregeometric shell of an arithmetically Cohen-Macaulayclosed subscheme V. Then the subscheme $W$ is also an arithmetically Cohen-Macaulay subscheme. (N.B. If $dim(W)=dim(V)$ , then this is obviously true by Auslander-Buchsbaum formula on the depths and the homological dimensions. Thus, in this case, $e.g$ . if $dim(V)>0,$
then the case:never occurs.)$W=V\mathrm{I}\mathrm{I}\{1 pt.\}$(0.1.2). ThenAssume that a closed subscheme $W$ is a pregeometric shell of a clos $ed$ subvariety
the subscheme $W$ is also a variety, namely reduced and irreducible. (N.B. When the subscheme$W$ is a schemeof codimcnsion one, this is true. cf. Lemma 3.2, Proposition 2.5 (2.5.1)$V\subseteq P$$)$(0.1.3)satand $W$ of positive dimension. Assume that the subvariety
Take closed subvarieties(depth $\geq 2$ ) condition, namely the natural map $H^{0}(P, O_{P}(m))\mathrm{e}$isfies the arithmetic$H^{0}(V, O_{V}(m))$ is surjective
for every integer $m$ . If $W$ is a pregeometric shell of , then on$:=dim(V)+deg(O_{V}(1))-h^{0}(V, O_{V}(1))$ ; (cf. [6]), the intheir -genera
equality:$V$$V$$D_{2}$$V$$\triangle$$(\mathrm{e}.\mathrm{g}.

\triangle(V, \mathrm{O}_{V} (1))$$\triangle(\mathrm{I}/, O_{V}(1))$$\geq\triangle(W, O_{W}(1))$

As a typical case, if the polarized manifold ( , Oy (1)) is arithmetically no rmal andis a hypersurface cut of the polarized manifold $(W, O_{W}(1))$ , then $W$ is a $PG$ -shell of
andthis inequality is obviously true. On the other hand, if we assume thatis non-degeneratearithmetically Buchsbaum, and $W$ is a hypersurface, then the result 71 on spec $ial$ cases ofEisenbud-Goto conjecture shows that this claim is also true.)holds, ( $e.g$ .

$V$

$V$

$V$

$7f$

(0.1.4)

Take a closed subvariety $W$ , a vector bundle $E$ on $W$ , a section $\sigma\in\Gamma(W, E)$ , and put: $V=Z(\sigma)$ .
condition; (b) the
Assume $:(a)$ the subscheme
is a variety and satisfies the arithmetic
restricted section $\sigma|_{Reg(W)}\in\Gamma(Reg(W), E|_{Reg(W)})$ is transverse to the zero section on the open
set $Reg(W),\cdot(c)W$ is a pregeometric shell ofand $Reg(V)\subseteq Reg(W)$ . Then the bundle
is a $nef$ bundle (N.B. If we do not assume the arithmetic
condition on , we havecounter-example, cf. [21]. On the other hand, if $W=P$ and rank(E) $=2,$ this claim is true).
$V$$D_{2}$

$V$

$E$

$V$

$D_{2}$

Once we fix a closed scheme

$V$

, its

$\mathrm{P}\mathrm{G}$$a$-shells generally exist infinitely many, but by the elementary

properties of-shells (cf. Proposition 2.5 (2.5.4)), they are bounded by an algebraic family of finite
components, which suggests the theoretical possibility of classifying its-shells completely. Thus, to$\mathrm{P}\mathrm{G}$$\mathrm{P}\mathrm{G}$-shells of the variety
find evidences for these conjectures, what we should do first is to classify all theof -genus zero.
-shells, we can easily reduced this problem
Applying an elementary property (cf. Lemma 2.11) of
to the case that the variety is a non-degenerate rational normal curve of degree $d\geq 3.$ Then the easiest
but non-trivial case is the case : $d=4.$ This case also relates with a Veronese surface which is the only
heretical (and the most interesting) case in the list of the varieties with -genus zero.
In this article, with a help of a classical result on varieties of minimal degree (cf. [12]), or of Fujita’s
modern theory on polarized varieties by using -genera (cf. [6]), using rather primitive and classicalmethods, we study these two cases as a first step toward those conjectures. After the classifications of-shells, wc see that all the claims above are affirmative in these two cases.
The author would like to express his deep gratitude to Prof. M. Green for inviting him to UCLA,
which brought hirn calm days and a nice environment for research, to Prof. T. Ashikaga for much incentive
for this classification (cf. [1]), and to Prof. K. Konno for giving a chance of the talk to the author.
$\mathrm{P}\mathrm{G}$$V$$\triangle$$\mathrm{P}\mathrm{G}$$V$$\triangle$$\triangle$$\mathrm{P}\mathrm{G}$128\S 1 Main Results.
To give an overview, let us summarize our main results as two main theorems. We should emphasize
again here that our “schemes” of course may have a non-equidimensional component or a non-reduced
structure. For precise definitions on terminology, see the next section 52.
-shells of a rational normal quartic curve. We
The first main theorem is on the classification ofwill give its full proof in the latter part of this article.
$\mathrm{P}\mathrm{G}$Main Theorem 1.1 Let $X$ be a non-degenerate non-singular rational quartic curve in a 4-th projectivewith the tautological line bundle $O_{P}(1)=O_{P}(H)$ , and a closed subscheme $W\subseteq Pa$
space:
pregeometric shell of $X$ with coclirn(W, ) $>0.$ Then the scheme $W$ is one of the following cases.
$P=\mathrm{P}^{4}(\mathbb{C})$$P$(1.1.1), thenbe a divisor ofand an irreducible and reduced quadric hypersurface
If codim
4,
5(
3,
non-singular).
rank
or
.
None
of
of these quadric hypersurfaces is a FG-shell(cf.
DefinitiOn2.4) of .
If codim(W) , then the scheme is irreducible and reduced. The polarized variety
is of -genus zero (of minimal degree). More precisely, the surface
is a projective cone of. In the-singular twisted cubic curve or one point blow-up of a projective planeaformer case, the curve passes through the vertex of the cone, arid is not a Cartier divisor
Hence there is no shell frame ofis embedded byin W. In the latter case, the variety
linear system coming from the conies passing the center of blow-up, and the curve
is a $nef$(Cartier) divisor on(thus a unique shell frame ( $O_{W}(X)$ ,) exists), whichomes fromsingular irreducible and reduced cubic plane curve passing through the center of blow up
or from an irreducible conic which does not pass through the center of blow-up (more precisely,
cf. Proposition 3.6 and ).
If codim(W) , then the scheme exactly coincides rryith .
$W$$(W)=\mathit{1}$$P$$\iota.e$$X$(1.1.2)$W$$=\mathit{2}$$(\mathrm{W}, O_{W}(1))$$W$$\triangle$$nor\iota$$\mathrm{P}^{2}(\mathbb{C})$

$a$

$X$

$W$

$X$

$a$

$X$

$W$

$co$

$\sigma x$$a$$doubl\mathrm{s}/$$S\mathit{4}\cdot$(1.1.3)$X$$W$$=\mathit{3}$Including the trivial case: codim(W) $=0$ , i.e. $W=P$ , a polarized scheme $(W, O_{W}(1))$ which ispregeometric shell of the curve $X$ is always a variety of -genus zero (of minimal degree), and therefore
an anthrnetically Cohen-Macaulay variety.

$a$$\triangle$

Based on Main Theoreml.l , applying Lemma2.11, we can get the next main theorem on a classificationof-shells of a Veronese surface which is the only heretical case in the classification on the varieties of-genus zero. In this article, we omit its proof because of the page limit. For its full proof, see [24].
$\mathrm{P}\mathrm{G}$$\triangle$

byMain Theorem 1.2 Let $X$ be a Veronese surface, namely the image of a projective $pla$ ne, in a 5-th projective
Veronesean embedding of degree 2 given by a complete linear system :“space:
with the tautological line bundle $O_{P}(1)=O_{P}(H)$ , and a closed subscheme $W\subseteq Pa$
pregeometric shell of $X$ with codim(W, ) $>0.$ Then the scheme $W$ is one of the following cases.
$\mathrm{P}^{2}(\mathbb{C})$$a$$|\mathrm{O}\mathrm{p}\circ-(\mathrm{C})(2)|$$P=\mathrm{P}$$(\mathbb{C})$$P$(1.2.1)

If codim(W) , then be a divisor of and an irreducible arid reduced quadric hypersurface
None of these quadric hypersurfaces is a $FG$ shell of .
If codim(W) , then the scheme is reducible and reduced. The polarized variety $(W, Ow(1))$
is only
is of gervus zero (of minimal degree). More precisely, The singular locus Sing(W) of
15):
variety(cf.
Lemma2.
one point , which is included by the surface V. The one point chordal$Cd(v_{0}, V)$notdoeshyperplain
whichof the surface coincides with the variety W. Take a$W$$=\mathit{1}$

$P$

$V$

(1.2.2)$W$$=\mathit{2}$

$i$

$W$

$\triangle_{-}$$v_{0}$$H$$V$

transversely. Then, the hyperplain cut $W\cap H$ isand intersects the variety.
Thus, the variety $W$ is also a projective cone of
one point blow up ofpass the point
isomorphic to
one point blout up of$v_{0}$$a$$W$$\mathrm{P}^{2}(\mathbb{C})$$\mathrm{P}^{2}(\mathbb{C})$

.

$a$

30

(1.2.3)

If codim(W)$=d\mathit{6}$, then the scheme $W$ exactly coincides with $X$ .

Including the trivial case: codim(W) $=0,$ $i.e$ . $W=P,$ a polarized scheme $(W, O_{W}(1))$ which ispregeometric shell of the curve $X$ is always a variety of -genus zero (of minimal degree), and therefore
an arithmetically Cohen-Macaulay variety.

$a$$\triangle$

Remark 1.3 With respect to a polarized variety $(V, L)$ . namely a pair of a projective variety
andan ample line bundle
on , the main concern of Fujita theory or of the theory of -genus is in
the geometric analysis of the embedding into a weighted projective space associated to the line bundle
instead of the embedding into a usual projective space by the linear system. However, if we restrict
ourselves to the case that the line bundle
is simply generated, or the variety
is embedded into
usual projective space with the “arithmetic“conditeon, Fujita’s theory can be applied directly to our
problems. Thus, in this article, if ate say that the subvariety
is a variety of -genus
zero, it means that the pair ( , Ox (1) ) is a variety of -genus zero in the sense of Fujita ’s theory andthe va riety
is embedded into $P$ by the comlete linear system Ox (1)(and therefore the variety
is non-degenerate in ), where the simple generation of the (very) ample line bundle $O_{P}(1)|$ is always
guaranteed by Fujita’s theory for the variety of -genus zero (cf.
).
$V$

$L$

$V$

$\triangle$$\prime s$

$L$

$|L|$

$L$

$V$

$a$

$D_{2}$$V\subset P=\mathrm{P}^{N}(\mathbb{C})$$V$$\triangle$$\triangle$$|\mathrm{v}$

$V$

$|$

$|$

$V$

$V$

$|$

$P$

$V$

$\triangle$$[\mathit{6}f$j2 Preliminaries.
To avoid needless confusions, let us confirm our notation used in this article.
Notation and Conventions 2.1 In this paper, we use the terminology of [8] without mentioned, andalways admit the conventions and use the notation below for simplicity.
(2.1.1)

Every object under consideration is defined over the field of complex numbers C. We will work in
the category of algebraic schemes over and algebraically holomorphic morphisms (or rationalmaps) or in the categories of coherent sheaves and their ( -linear) homomorphisms othe rwise
mentioned.
$\mathbb{C}$$O$(2.1.2)

Let us take a complex projective scheme $X$ of dimension and one of its embeddings :. The sheaf of ideals defining $j(X)$ in $P$ and the conormal sheaf are denoted by, respectively. Taking
and-basis $\{Z_{0}, \ldots, Z_{N}\}$ of $H^{0}(P$, Ox (1 ) . Then we
put.
$j$

$n$

$P=\mathrm{P}^{N}(\mathbb{C})$$X\mathrm{c}arrow$$I_{X}$$a\mathbb{C}$$N_{X/P}^{\vee}=I_{X}/I_{X}^{2}$$\cdot$$S$$:=$$\oplus H^{0}(P, O_{P}(m))\mathrm{Y}$$\mathbb{C}[Z_{0}, \ldots, Z_{N}]$$m\geq 0$$s_{+}$$\overline{R_{X}}$:$:=$$= \bigoplus_{m>0}H^{0}(P, O_{P}(m))\cong(Z_{0}, \ldots, Z_{N})\mathbb{C}[Z_{0}$,... ’$Z_{N}]$$\oplus H^{0}(X, O_{X}(m))$$m\geq 0$$\Pi_{X}$$R_{X}$

$:=$

$:=$

$\bigoplus_{m\geq 0}H^{0}(P, I_{X}(m))$$Im[Sarrow\overline{R_{X}}]\cong$

S/I[X.

In this case, the induced ample line bundle $j’ O\mathrm{O}(1)=j^{*}O_{P}(H)$ is denoted by $O_{X}(1)$ or $O_{X}(H)$ .
, then its tautological ample line bundle
If the scheme $X$ itself is another projective spaceis denoted simply by $O(1)$ without any letter to specify the space, otherwise mentioned
explicitly. Thus, if we consider the -th Veronesean embedding :for $X=$, then there is an identification of the line bundles $O_{X}(H)=Ox$ $(1)=O(d)$ .
$\mathrm{P}^{n}(\mathbb{C})$$o_{\mathrm{r}P^{n}(\mathrm{C})}(1)$$d$$\mathrm{P}^{n}(\mathbb{C})$$j$$X\mathrm{c}arrow P=\mathrm{P}^{N}(\mathbb{C})$131(2.1.3)For a graded -module $E$ , the symbol $E(m)$ denotes the degree $m$ part of $E$ , namely.

$S$$E= \bigoplus_{m\in \mathrm{Z}}E_{(m)}$The degree shift: $E(d)$ of the module $E$ means that $(E(d))(m):=$ E{. Affine sheafication of
the -module $E$ ($i.e$ . a canonically constructed $O_{Spec(S)}$ -module from a -module ) is denotedbyand projective sheafication of the graded -module $E(i.e.$ a canonically constructed$o_{Proj(S)}$ -module from a graded, respectively.
-module ) is denoted by$\mathrm{d})$

$S$

$S$

$S$

$E^{\sim}$$E^{(\sim)}$$S$(2.1.4)For a coherent sheaf
to be:$F$

on a closed subscheme$V\subseteq P,$

we set the Hilbert polynomial of the sheaf

$F$

$N$

$A_{F}(m):=$ $\mathrm{x}(F(m))=E(-1)^{q}\dim(H^{q}(V, F(m)))$

.

$q=0$

In case of $F=Oy,$ we denote its Hilbert polynomial as $A_{V}(m)$ instead of $Ao_{v}(m)$ . Moreover,, we simply write $A_{k}(m)$ for. (Precise properties of
if
Hilbert polynomials are referred to$(V, O_{V}(1))=(\mathrm{P}^{k}(\mathbb{C}), O_{\mathrm{I}\mathrm{P}^{k}(\mathrm{C})}(1))$$\mathrm{J}_{\mathrm{P}^{k}(\mathbb{C})}(m)$$[\mathit{1}\mathit{1}^{l},)$(2.1.5)For a real valued function $f(x)$ defined on the field of real numbers (or on the ring of rational
integers ), we define the (first) (backward) difference function to be:f(x) $-f(x-1)$
and the -th difference function to be $(\nabla^{k}f)(x):=(\nabla(\nabla^{k-1}f))(x)$ for a positive integer.
$(\nabla^{0}f)(x):=f(x))$ . The operator
is called (backward) difference operator.
$\mathbb{R}$$\mathbb{Z}$$(\mathrm{V}\mathrm{f})(\mathrm{x}):=$

$k$

$k$

$(\mathrm{N}.\mathrm{D}$$\nabla$(2.1.6)

Let

$B$

set :

curve
Opi

be a projective line$\pi$

$C_{e}$:$\mathrm{P}^{1}(\mathbb{C})$. For a non-negative integer (or to include $e=1$ ), we( . . a rational ruled surface of degree ),$e$$\Sigma_{e}:=\mathrm{P}(O_{\mathrm{P}^{1}(\mathrm{C})}\oplus o_{\mathrm{F}^{1}}(\mathrm{C}) (-e))arrow B,$

$i$

$.e$

$e$

$e$

$a$

in the surface
to be a tautological -ample divisor dete rmined by the vector bundle$(-e)$ and a curve
to be the fibre of the morphism .
$\Sigma_{\mathrm{e}}$

$\pi$

$f$

$(\mathrm{c})\oplus O_{\mathrm{P}^{1}}(\mathrm{C})$$\pi$

We will need a classical and well-known result on the Picard group of a rational ruled surface (cf. [8]).

Lemma 2.2 Under the circumstances of (2.1.6), let us consider a rational ruled surface. Then the
curves: $\{f, C_{e}\}$ forrn -free basis of the Picard group $Pic(Ee)$ of the surface
and have the following$\Sigma_{e}$$\mathbb{Z}$$\Sigma_{e}$properties.

(2.2.1)

The intersection numbers are : $c_{e}^{2}=-e$ $\leq 0;f^{2}=0;$ and .Ce $=1.$(2.2.2)For integers$\mathrm{f}$$u$artd ,$v$

the chvisor $uf+vCe$ is very ample(2.2.3)For integers$u$$\Leftrightarrow the$$d\iota vis$or $uf+vC_{e}$ is ample 9 ” $v>0$ , $u>ve$ ”

and ,
$v$

the linear system
contains an irreducible-srngular curve
contains an irreducible curve 9 ”$(u, v)$ $=(1,0)$ or $(0, 1))$. or $v>0$ , $u>ve$ ; or $e>0,$ $?’>0,$ $u=ve$ ”.
$|uf+\uparrow’ C_{e}|$(2.2.4)$\Leftrightarrow it$$\tau\iota \mathit{0}r\iota$

On the canonical divisor of the surface$\Sigma_{e}$, we have$K_{\Sigma_{e}}=(-2-e)f+(-2)C_{e}$.

Since in our classification, we have to consider non-reduced and non-equidimensional schemes, need an
accurate handling on their nilpotent structures, and can not reduce our problems to the case of reduccd
structures, we have to pay special attention to “primary (or irreducible )” decompositions of schemes.

132

Definition 2.3 For a Noetherian scheme $W$ and finite number of its closed subschemes
schemetheoretic union, namely the kernel sheaf:
is defined by an ideal sheaf $I_{Y}:=$ I
On the other hand, since there is no uniqueness of primary decomposition of ideals
in Noetherian rings, once if an arbitrary Noetherian scheme
which may have a nilpotent structure isgiven first, then it is not so trivial to find its closed subschemes
such that each topological spacecoincides with Y. Thusof the subscheme is irreducible and the scheme theoretic union
we will restrict ourselves to the case that the scheme is a closed subscheme ofProj(S)and will define its “primary decomposition” and “irreducible decomposition” (not uniquely). For the h0mogeneous idealof , ) have a homogeneous primary decomposition in the shortest representation:, and we set 7.
, $V_{s}:=(Supp(Op/J_{5}), O_{P}/J_{\mathrm{s}})$ for $s=1$ ,’, . It is easy to seethat their scheme theoretic union
coincides with Y. We call this finite set of closed subschemes.
as a primary decomposition of Y. Next we pick up all the minimal prime ideals of . Then
we may assume thatare the primary ideals associated to these minimal prime ideals,
respectively As is well-known, only these primary ideals:(resp. only these “maximal” primary
component subschemes :(resp. by the subscheme $Y$ ).
are determined uniquely by the ideal
Now, for $i=1,$ . . . ,set a subscheme
to be the scheme theoretic union:$\{V_{s}\}_{s=1r}^{k^{\mathrm{r}}}$$Y= \bigcup_{s=1}^{k}V_{s}$$s=1k$$I_{V_{\mathrm{s}^{\neg}}}$$Ker[O_{P}arrow\oplus_{s=1}^{k-}O_{V_{s}}]$$Y$$\{V_{s}\}_{s=1}^{k}$$|V_{s}|$$\bigcup_{s=1}^{k}.V_{s}$$V_{s}$$Y$$Y$$\mathrm{I}_{Y}$$\zeta 1’=$$u$

$P=\mathrm{P}^{N}(\mathbb{C})=$$e$$:=(\mathrm{J}_{s})^{(\sim)}$$s=1k^{\wedge}\mathrm{J}_{s}$$\ldots$$\bigcup_{s=1}^{k}$

$k$

$V$

$\{V_{s}\}_{s=1}^{k}$$\mathrm{I}_{Y}$$\{\mathrm{J}_{i}\}_{i=1}^{t}(t\leq k)$$\{\mathrm{J}_{\mathrm{z}}\}_{i=1}^{t}$$\{V_{i}\}_{i=1}^{t})$$t$’$\mathrm{I}_{Y}$$U_{i}$$l\mathit{1}C\mathit{2}$$U_{i}:=V_{s}\underline{\mathrm{C}}V_{1}\cup V_{s}$.

Then we call the finite set of closed subschemes
as an irreducible decomposition of Y. We
should make a remark that the homogeneous ideal
does not have the irrelevant maximal idealas
an associated prime by its definition, the homogeneous ideal
coincides with the ideal , and that thehomogeneous ideal
does with the ideal:$\{U_{i}\}_{\mathrm{z}=1}^{t}$$S_{+}$$\lrcorner \mathrm{v}_{Y}$$J_{s}$$\mathrm{I}_{V_{\mathit{8}}}$$\mathrm{I}_{U_{l}}$$\cap$$\mathrm{J}_{6}$.

$\sqrt{\mathrm{J}_{\mathrm{t}}}\subseteq\sqrt{\mathrm{J}_{\mathit{3}}}$

If $dim(Y)=n$ , then a primary componentof dimension$V_{6}$$n$(resp. an irreducible component

dimension ) is called a main pri mary component (resp. a main irreducible$n$$U_{i}$

of$\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}/$Now let us recall our key concepts for studying the geometric structures of projective embeddings,-shell”) ) of them were introduced first in [19]. On -shells, we havethe first two : ( “shell andslightly modified its definition from $V\subseteq Reg(W)$ in [19] to $Reg(V)\subseteq Reg(W)$ . By reconsidering classicalexamples in Complex Projective Geometry including varieties of minimal degree, with standing on thisnew point of view, we can find many good actual examples for these concepts in a number of works suchas [13], [15], [5], [7], [16], [6], [3] and so on.
($\mathrm{P}\mathrm{G}$$‘ \mathrm{G}$$\mathrm{G}$Definition 2.4 (shells and cores) Let
which satisfy
and $W$ be closed subschemes of(namely the inclusion of the defining ideal sheaves:in the structure sheaf
of $Pj$ In
$W$
case,
this
the subscheme
is called simply an intermediate ambient scheme of ) If the natural map:
$V$

$P=\mathrm{P}^{N}(\mathbb{C})$$V\subseteq W$$O_{P}$$I_{V}\supseteq I_{W}$$V$$\mu_{q}$: $Tor_{q}^{S}(R_{W}, S/S_{+})arrow Tor_{q}^{S}(R_{V}, S/S_{+})$

is injective for every integer $q\geq 0$ (abbr.
shell (abbr.
-shell,) of $V$ and that $V$ isinjectivity condition” ), we say that $W$ is pregeometric-core) of W. Moreover, if
pregeometric core (abbr.
and $W$ are closed subvarieties and the regular locus $Reg(W)$ of $W$ contains $Reg(V)$ , we say that $W$ isgeometric shell (abbr. -shell) ofandis geometric core (abbr. -core) of W. Furthermore, we
assume that (a) there exist a vector bundle $E$ on $W$ and a global section $\sigma\in\Gamma(W, E);(b)$ the zero locus(the scheme structure of) coincides
is defined by an ideal sheaf: $I_{Z(\sigma)}:=Im[$ :$” \mathrm{T}\mathrm{o}\mathrm{r}$$\mathrm{P}\mathrm{G}$

$a$

$V$

$\mathrm{P}\mathrm{G}$

$a$

$a$

$\mathrm{G}$$Z(\sigma)$$V$$\mathrm{Z}(\mathrm{a})$

$V$

$a$

$\mathrm{G}$$y$$E^{\vee}arrow O_{P}]$133

including their scheme structures; (c) the restricted section $\sigma|_{Reg(W)}\in$ Reg(W), $E|_{Fteg(W)})$ iswithtransverse to the zero section on $Reg(W)$ , then we say that $W$ is framed geometric shell (abbr. FGshell of and is framed geometric core (abbr.
core of W. In this case, the pairis called
shell frame of in W. For the subscheme , the total space $P$ and itself are called trivial PG-shells(or trivial -shells if is a variety).
$V$

$a$

$V$

$V$

$V$

$a$

$(E, \sigma)$$\mathrm{F}\mathrm{G}$

$a$

$V$

$V$

$V$

$\mathrm{G}$Let us recall a proposition on several elementary properties of-shells from [20]. The outline of
proof is referred to [20] (on the properties of -shells related to their restricted syzygy bundles andinfinitesimal syzygy bundles, which are not presented here, see [22] .
$\mathrm{P}\mathrm{G}$$\mathrm{G}$$)$Proposition 2.5 Let(2.5.1)

If
of

$V$

and $W$ be closed subschemes of$P=\mathrm{P}^{N}(\mathbb{C})$which satisfy$V\subseteq W.$

is a hypersurface ( . a divisor of $P$; cf. also Lemma 3.2), then $W$ is a pregeometric shell$V$
if and only if the equation of $W$ is a member of minimal generators of the homogeneous
ideal
of $V$ .
$W$$i.$ $e$$\mathrm{I}_{V}$(2.5.2)

Assume that the subscheme is a complete intersection. Then the scheme $W$ is a pregeometricshell of if and only if the subscheme $W$ is defined by a part of minimal generators of .
$V$

$V$

(2.5.3)$\mathrm{I}_{V}$Take a closed schemesuch that
W. Assume that $W$ is a pregeometric shell$W$
is also a pregeometric shell of Y. In particular, the subscheme $W$ is alsoof V. Thenpregeometric shell of the $m$ -th infinitesimal $Jv\cdot\cdot,ighborhood$ $Y=(V/W)_{(m)}$ of
in $W$ , ?1) here$Y$$V\subseteq Y\subseteq$

$a$

$V$

$(V/W)_{(m)}=(|V|, O_{W}/I_{1\nearrow/W}^{m+1})$(2.5.4).

Fix the subscheme
Then all non-trivial pregeometric shells of formof codim(V, )non empty algebraic family of finite components (N.B. The family of all non-trivial -shells of
may be empty even if itself is a smooth variety).
$V$

$P$

$V$

$\geq 2.$

$G$

$V$

(2.5.5)If$V$

is a pregeometric shell of , then we have an inequality: arith.depth(V)arith.depth(W)on their arithmetic depths. In particular, if $thc$ natural restriction map $H^{0}(P, O_{P}(m))$$H^{0}(V, O_{V}(m))$ is surjective
for all integers $m(i.e. R_{V}=\overline{R_{V}})$, then the natural restriction map$H^{0}(P, O_{P}(m))arrow H^{0}(W, O_{W}(m))$ is also surjective
for all integers $m(i.e.R_{W}=\overline{R_{W}})$ . In otherwords, the arithmetic
condition is inherited from pregeometric cores to their pregeometricshells.
$W$$V$$\leq$$arrow$$D_{2}$(2.5.6)

If the subscheme(2.5.7)

Assume that there exist hypersurfaces,,,... ,
in with homogeneous equations$V=W\cap
D_{1}$
degree
,
,
,
,
.
.
.
:(a)
respectively,’
satisfying
theconditions
andof$D_{t}(t= 1, \cdots, r)$(b) $H^{0}(W_{t}, O_{W_{\mathrm{t}}})=\mathbb{C}(t=0, \cdots, r)$, where $0:=W$ and; (c) the homogeneous equations,-regular sequence, namely the sequence :form an$W$ is a pregeometric shellthen weof the subscheme with arith.depth(V)aninequality
haveon their CastelnuovO-Mumford regularity(cf. [4]): $reg^{CaM}(V)\geq reg^{CaM}(W)$ .
$V$$\cdots$$D_{r}$$D_{1}$$r$$m_{1}$$\geq 2,$$P$

$F_{1}$$O_{W}$$\cdots F_{r}$,$0arrow O_{W_{\mathrm{t}-1}}(-m_{t})arrow\cross F$(2.5.8)$\cdots$$F_{r}$$W_{t}:=W\cap D_{1}\cap\ldots \mathrm{q}$$\mathrm{i}$$F_{1}$

is exact for $t=1,$$\cdots$$\cap D_{r}$$m_{r}$,$r$. If arith.depth(V)$\geq 2,$$\mathit{0}_{W_{\mathrm{t}-1}}$,

then $W$ is a pregeometric shell.

Assume that the subscheme
is non-degenerate, namely no hyperplane contains V. If $W$ has
a 2-linear resolution, . . the homogeneous coordinate ringof $W$ has a minimal S-free. .
resolution of the form : $0arrow R_{W}7$ $Sarrow$ Fi $(-2)arrow F_{2}(-3)arrow\cdots\vdash$ $Fp(-p ・1)$$W$
iswhere $F_{u}(v)$ denotes (&S(v) :a direct sum of several copies of with degree shift, thena pregeometric shell of V. (cf. also Corollary 3.7)
$V$

$i$

$e$

$R_{W}$

$+-$

$S$

$v$

’

134

Remark 2.6 Related to Proposition 2.5 there were two minor mistakes in the claims of [20]. The
first one was in the old version of the claim (2.5.6). The author had failed to attach the condition:arith.depth(V)by a misprint, which is corrected in the improved version of [.AGfOOOl 004.
The second one was a failure to attach the condition: $H^{0}(W_{t}, O_{W_{\mathrm{t}}})=\mathbb{C}(t=0, \cdots, r)$ of the claim
2.5, 7). Without this condition, the claim (2.5.7) is not true in general. A counterexample is given by
the next example.
$\geq 2$$\mathit{2}\mathit{0}f:math$$(N\geq 4)$ . take a smooth hypersuface
Example 2.7 In $P$ If
of degree $m_{0}\geq 2,$ a line
, namely
which intersects the hypersuface
transversely, and define a closed subscheme $W$ to be
by a sheaf of ideals :. Now we take a smooth hypersufaceof degree $m_{1}\geq 2$transversely, and satisfies $D_{1}\cap D_{0}\cap L_{0}=\phi$ . Then
which intersects the hypersuface
arid the line
$W_{1}=W\cap
D_{1}=(D_{0}\cap
D_{1})\cup${pi,
we see thatis a set of finite}, where the set
points :. Last we choose a smooth hypersu factof degree $m_{2}\geq 2$ with two conditions:. The scheme’;
transversely, and put $V:=W\cap D_{1}$intersects
coincides with. Then we see easily that arith.depth(V)the trvo hypersurfaces. Since theand-regular sequence, $H^{0}(O_{W})\cong H^{0}(O_{V})\cong \mathbb{C}$ , butform anis a complete intersection, its pregeometric shells are only complete intersections. However,variety
the scheme $W$ is obviously not a complete intersection, and therefore is not a pregeometric shell of V. In
this case, it is easy to check that arith.depth(W)by seei $ngh^{1}(O_{W})=m_{0}-1$
$N$

$=$

$D_{0}\subset P$$(\mathbb{C})$$L\circ$$D_{0}\cup L_{0}$$D_{0}$$D_{1}\subset P$$I_{W}:=I_{D_{0}}\cap I_{L_{0}}\subset \mathit{0}_{P}$$D_{0}$$L_{0}$$\{\mathrm{p}\mathrm{i}, \cdots,p_{m_{1}}\}$$\cdots,p_{m_{1}}$$D_{1}\cap L_{0}$$D_{2}\subset P$$D_{2}\cap\{p_{1}, \cdots, p_{m_{1}}\}$
$V$

$=$

$D_{2}$”$D_{0}\cap D_{1}$$D_{2}$$D_{0}|\gamma D_{1}\cap D_{2}$$O_{W}$$D_{2}$$D_{1}$$\geq 2,$$H^{0}(O_{W_{1}})\cong \mathbb{C}^{\oplus m_{1}+1}$$V$$\geq 1.$$\leq 2$

Relating to Remark(2.6), we summarize an easy result on homogeneous coordinate rings of closed
subschemas, which is well-known in the case of closed subvarieties.

Proposition 2.8 Let $W\subseteq P=\mathrm{P}^{N}(\mathbb{C})(N\geq 2)$ be a closed subscheme satisfying $H^{0}(O_{W})=\mathbb{C}$ ,$D\subset P$ a hypersurface with a homogeneous equation $F$of degree . Assume that the equation $F$ of$D$ isanla element, namely the multiplication :: $O_{W}(-m_{0})arrow Ow$ is injective, and thatarith.depth(W) $+1.$
Then, $R_{W\cap D}=R_{W}/F.R_{W}$ , and arith.depth(W) arith.depth(W)$m_{0}$$O1\mathrm{t}$-$\cross F$$r$$r\cdot egu$$\cap D$$\geq 2.$$\cap D$$=$

Before starting its proof, let us recall an easy fact from (5.2)Lemma of [18].
be a closed subscheme satisfying $\dim(W)\geq 1$ and $H^{0}(Ow)=\mathbb{C}$ , andthe restriction bundle of the tautological ample line bundle $O_{P}(1)$ . Then $H^{0}(O_{W}(-m))=0$ for
any positive integer $m$ .

Lemma 2.9 Let$W\subseteq P=\mathrm{P}^{N}(\mathbb{C},)$$o_{\nu v}(1)$Proof of Proposition (2.8) Since we assume arith.depth(W)
we have the surjectivity of therestriction map: $H^{0}(P, O_{P}(m))arrow H^{0}(W\cap D, O_{W\cap D}(m))$ for any integer. Let us show first thatarith.depth(W)namely the surjectivity of the restriction map : $H^{0}(P, O_{P}(m))arrow H^{0}(W, O_{W}(m))$
for any integer
by induction on $m$ . By the assumtion, we can apply Lemma(2.9) to the scheme$W$ and may assume that the restriction map:)
) is surjective for any( $W$, Ow( $P$, Op$m’<m$
integerwithas an induction hypothesis. We apply the snake lemma to the diagram:$\cap D$$\geq 2,$$m\in \mathbb{Z}$$\geq 2,$$m\in \mathbb{Z}$$H^{0}$$(\mathrm{m})$$arrow H^{0}$$\{\mathrm{m}’$$)$$rn’\in \mathbb{Z}$

0

0
$|$

$\uparrow$

0 $-arrow H^{0}(W, O_{W}(m-m_{0}))arrow \mathrm{x}FH^{0}$ ( $W$, Op$\uparrow|$

0 $-arrow H^{0}(P, O_{J^{\mathit{3}}}(m-m_{0}))$$(\mathrm{m})$) $arrow H^{0}(W\cap D, O_{W\cap D}(m))arrow 0$$\uparrow$$arrow \mathrm{x}FH^{0}$( $P$, Op(m)) $arrow$$\uparrow$$H^{0}(D, O_{D}(m))$$arrow 0$

135$(P, O_{P}(m)$ $arrow H^{0}(W, O_{W}(m))$ (This argument is theand see the surjectivity of the restriction map:
same to apply Nakayama’s lemma to the -module $\oplus_{m}H^{0}(W, O_{W}(m))$ based on the finite generation ofthis module asserted by Lemma(2.9) . Then, we again apply the snake lemma to the diagram:$H^{0}$

$)$

$S$

$)$

0

0

$\uparrow$$\uparrow$$0arrow H^{0}(W, O_{W}(m-m_{0}))arrow\cross FH^{0}$ ( $W$, Oy$\uparrow$$(\mathrm{m})$)$arrow H^{0}(W\cap D, O_{W\cap D}(m))$$\mathrm{t}$$0arrow$$arrow 0$$\uparrow$$arrow\cross F$$arrow$$(R_{W})_{(m-m\mathrm{o})}\uparrow$$(R_{W})_{(m)}\uparrow$00$arrow 0$$(R_{W}/F.R_{W})_{(m)}$and see that $RwnD=Rw/F.Rw$ , which implies that arith.depth(W)$=$arith.depth(W”$D$) $+1.$

Corollary 2.10 Let
and It be closed subschemes of $P=$ IF
which satisfy $V\subseteq W$ . Assume that
there exist hypersurfaces.,
in $P$ with homogeneous equations ,,of degree , , ,respectively, and satisfying the conditions :(a) $V=W$ ”; (b) $H^{0}(W_{t}, O_{W_{L}})=\mathbb{C}(t=$, . ’ ), where $W\circ:=W$ and $W_{t}:=W\cap D_{1}$ ”). . . $\cap D_{t}(t= 1, \cdot\cdot, r)$ ; (c) the homogeneous equations,-regular sequence. If arith.depth(V)then $R_{W_{\ell}}=R_{W}/(F_{1}, \cdots , F,)R_{W}$ , theform ansequence ,,sequence,-regularandarith.depth(W)arith.depth(V) $+r.$
form
$V$

$N(\mathbb{C})$$D_{1}\ldots$$r$$D_{r}$$F_{1}$$F_{r}$$\cdots$$\mathrm{m}\mathrm{i}$$m_{r}$$\cdot$$D_{1}\cap\ldots\cap D_{\tau}$$0$$F_{1}$$\cdot$$r$$\cdot$$\cdots$$O_{W}$$F_{\mathrm{r}}$$F_{1}$$\cdots$$\geq 2,$

$R_{W}$$F_{r}$$=$

By the similar argument, we can obtain the next helpful lemma for studying-shells of arithmetically
Cohen-Macaulay-cores. In these cases, this lemma guarantees that we can apply the hyperplain cut(or ” Apollonius” ) method to reduce the problems on higher dimensional-cores into those on PG-core$\mathrm{P}\mathrm{G}$$\mathrm{P}\mathrm{G}$$\mathrm{P}\mathrm{G}$curves.

Lemma 2.11 Let$V$

and $W$ be closed subschemes of$P=\mathrm{P}^{N}(\mathbb{C})$which satisfy $V\subseteq W.$(2.11.1) Assume that arith.depth(V)and the scheme $W$ is a pregeometric shell of $V$ in P. If we take$H\subsetP$
a hyperplain
with a linear equation $F$ which is anand-regular element, then the$H$schemeis a pregeometric shell of $V\cap H$ in the projective space $H\cong$ IF(or in $P$ ).
$\geq 9.$$O_{V}$$O_{W}$$W\cap \mathit{1}$$N-1(\mathbb{C})$(2.11.2) Suppose that $H^{0}(O_{V})=H^{0}(O_{W})=\mathbb{C}$ . Take a hyperplain $H\subset P$ with a linear equation $F$
which is anand-regular element. Assume that arith.depth(V $\cap H$ )and the scheme$W\cap H$ is a pregeometric shellof $V\cap H$ in the projectiue space $H$ (or in $P$ ). Therz the scheme$W$ is ( pregeo’rnet’ $ic$ shellof $V$ in $P$ .
$O_{V}$$O_{W}$$\geq 2$$s$

Proof. Let us suppose the assumption of the claim (2.11.1). By the definition, the ringscanandbe regarded as subrings of( , Oy (m)) and of( $W$, Oy (m)), respectively, which implies thatthe equation $F$ is a regular element for
and for. The claim (2.5.5) shows that arith.depth(W)arith.depth(V)which implies that $depth_{S_{+}}$ Rw/F.Rw, $\geq depth_{S_{+}}$ (RV/F.RV)
Thus we seethat $RvnH=R_{V}/F.R_{V}$ and $RwnH=R_{W}/F.R_{W}$ . By tensoring f
to the exact sequence:$R_{V}$$\oplus_{m}H^{0}$$V$$R_{W}$$\oplus_{m}H^{0}$$R_{V}$$R_{W}$$\geq$$\geq 2,$$\geq 1.$

$S$

$S_{+}$$0arrow R_{V}(-1)arrow\cross FR_{V}arrow RvnH$ $arrow 0,$

3El

we have an exact sequence:$Tor_{q}^{S}(R_{V}(-1), S/S_{+})\underline{\cross F}\rangle$$Tor_{q}^{S}(R_{V}, S/S_{+})$$arrow \mathit{7}$RVnH $S/S_{+}$ )$or_{q}^{S}$$arrow Tor_{q-1}^{S}(Rv(-1), S/S_{+})arrow \mathrm{x}FTor_{q-1}^{S}(R_{V}, S/S_{+})$ .

Since the modules: $Tor_{q}^{S}(R_{V}(-1), S/S_{+})(q\geq 0)$ are $S/S_{+}$ -modules, the multiplication by annihilates$(R_{V}(-1), S/S_{+})(q\geq 0)$ . After the similar argument on, we have an exact
the modules:
commutative diagram:$F$$R_{W}$$Tor_{q}^{S}$$0arrow$ 7 $or_{q}^{S}(R_{W}, S/S[perp])-arrow Tor_{q}^{S}(R_{W\cap H}, S/S_{+})arrow Tor_{q-1}^{S}(R_{W}(-1), S/S_{\tau})arrow 0$$\downarrow|\mu_{q-1}$$\downarrow\overline{\mu_{\mathrm{q}}}$$\downarrow|\mu_{q}$$0arrow Tor_{q}^{S}(R_{V}, S/S_{+})arrow Tor_{q}^{\mathit{8}}(R_{V\gamma H}, S/S_{+})arrow Tor_{q-1}^{S}(R_{V}(-1), S/S_{+})arrow 0.$

Then the assumption asserts the injectivity of the map, which brings theand of the map
injectivity of the map, or equivalently the scheme $W$ \cap $H$ is a-shell of $V\cap H$ in the projective
space $P$ .
Next let us suppose the assumption of the claim (2.11.2). Then, the claim (2.5.5) shows thatarith.depth(W $\cap H$ )arith.depth(W $\cap H$ )By Proposition (2.8), we see that $RVnH=R_{V}/F.Rv$ ,$=R_{W}/F$.
and the equation $F$ is a regular element for. By the similar
and forargument in the proof of the claim (2.11.1) above, we get an exact commutative diagram:$\mu_{q-1}$$\mu_{q}$$\mathrm{P}\mathrm{G}$$\overline{\mu_{q}}$$\geq 2.$$\geq$$R_{W\cap J}f$

$R_{W}$$R_{W}$$R_{V}$$0arrow Tor_{q}^{S}(R_{W}, S/S_{+})arrow Tor_{q}^{S}(R_{W\cap H}, S/S_{+})arrow Tor_{q-1}^{S}(Rw(-1), S/S_{+})arrow 0$$\downarrow\mu_{q}$$0arrow Tor_{q}^{S}$-Ry,$\mathrm{v}|^{\mu_{q-1}}|$$\downarrow^{1}\overline{\mu_{q}}$$S/S_{arrow}$)$arrow Tor_{q}^{S}.(R_{V\cap H}, S/S_{+})arrow Tor_{q-1}^{S}(R_{V}(-1), S/S_{+})arrow 0.$

In this case, the assumption asserts the injectivity of the, which induces the injectivity of the$V$ in. Thus we see that the scheme $W$ is amap-shell ofthe projective space $P$ .
Now let us show that the scheme $Y:=W\cap H$ is a-shell of the scheme $X:=V\cap H$ in $H$ if andonly if the scheme $Y:=W\eta$ $H$ is a-shell of the scheme X.— $V\cap$ $H$ in $P$ . We may assume that
$H=V_{+}(Z_{N})=$ Proj{T), where $T=S/Z_{N}.S\cong \mathbb{C}[Z_{0}, \cdots, Z_{N-1}]$ . Take a point
in $P$ which is notcontained in the hyperplain $H$ . Then the space $P$ can be considered as the projective conewithvertex , namely the ring can be considered as the polynomial ring $T[Z_{N}]$ , which is faithfully flatover the ring . Consider the projective cones
of $X$ and $\hat{Y}:=C_{p0}(Y)$ of $Y$ with the vertex$X=\hat{X}\cap
H$$Y=\hat{Y}$$H$ , and the equation$p0$ . Then $R_{\hat{X}}=R_{X}3\tau$, $R_{\hat{Y}}=R_{Y}\otimes_{T}S$ ,is a regular,element for the ringand for the ring
. Thus, starting from the minimal 7-free resolutions ofand of, we can construct the minimal 5-free resolutions ofand the minimal 5-freeand ofresolutions of $R_{\hat{X}\cap H}=R_{X}$ and of $R_{\hat{Y}\cap H}=R_{Y}$ , successively. We can also construct all the induced
homomorphisms of complexes for those minimal resolutions compatibly. Through these constructions,
we see that:$\mathrm{m}\mathrm{a}\mathrm{p}:\overline{\mu_{q}}$$\mathrm{P}\mathrm{G}$$\mu_{q}$$\mathrm{P}\mathrm{G}$$\mathrm{P}\mathrm{G}$$p_{0}$$C_{\mathrm{P}0}(H)$$S$$\mathrm{c}\mathrm{h}t^{1}$$p_{0}$$T$$\hat{X}:=C_{\mathrm{P}0}(X)$”$S$$R_{\hat{X}}$$Z_{N}$$R_{X}$$R_{\hat{Y}}$$R_{Y}$$R_{\hat{X}}$$R_{\hat{Y}}$$Tor_{q}^{S}(R_{W}, S/S_{+})\cong Tor_{q}^{T}(R_{W}, 7 /\mathrm{Y}_{+})\oplus T\mathrm{m}’-1(R_{W}, T/T_{+})$$\mathrm{t}^{\mu_{\mathrm{q}},S}$$\downarrow\mu_{q},\tau$$\downarrow^{\mu_{q}-1T}|$$Tor_{q}^{S}(R_{V}, S/S_{+})\cong Tor_{q}^{T}(R_{V}.T/T_{+})\oplus Tor_{q-1}^{T}(R_{V}, T/T_{+})$

which shows that all the maps:
which is what we wanted.

$\{\mu_{q},s\}_{q\geq 0}$

are injective if and only if all the maps:,$\{\mu_{q},\tau\}_{q\geq 0}$

are injective,I137

Related to Conjecture 0.1(0.1.1), we have a very easy result which srnplifies our argument later.
Proposition 2.12 Let
be a variety of -genus zero (cf. Remark 1.3) whose dimension
is , and a scheme $W\subset P$ a pregeometric shell of V. Assume that the scheme $W$ is arithmetically
Cohen-Macaulay and of (pure) dimension $m$ . Then, the scheme $W$ is also a variety of -genus zero.
$V\subset P=\mathrm{P}^{N}(\mathbb{C})$$\triangle$$n$$\triangle$

Proof. By the results of [6] and [5], our assumption implies that the homogeneous coordinate ringis Cohen-Macaulay and has a 2-linear minimal -free resolution. The minimal -free resolution of,. : $0\prec-R_{V}\mathrm{w}$ $\mathrm{F}_{V,0}=Sarrow \mathrm{F}_{V,1}=\oplus S(-2)arrow i$ , $2=\oplus S(-3)arrow\cdotsarrow \mathrm{F}_{V,p}=$is of the form:. . $arrow \mathrm{F}_{V,r}=\oplus S(-r ・1)$ , where $r=N-$ n. Since we assume that the scheme $W$$\oplus S(-p ・1)$
is arithmetically Cohen-Macaulay and is a-shell of $V$ , the homogeneous coordinate ringhas a$\mathrm{F}_{W,0}=Sarrow-\mathrm{F}_{W,1}=\oplus S(-\underline{9})+-\mathrm{F}_{W,2}=$,. : 07
minimal -free resolution of the form:. $\succ \mathrm{F}_{W,t}=\oplus S(- t-1)$ , where $t=N-m=r-(m-n)$ . Then we apply Theorem 4.1.15 of[2] and see that $e:=deg(V)=(1/r!)\Pi_{k=1}^{r}(k+1)=r+1$ and $e’:=$ deg(V) $=(1/t!)\Pi_{k=1}^{t}(k+1)=t+1.$
Let us show that the scheme $W$ is a variety. Take a main primary component (cf.Definition 2.3):
of$W$ which includes, give a reduced structure on, and put $\overline{e_{0}}:=deg((W_{0})_{red})$ . Since the scheme $W$
is arithmetically Cohen-Macaulay, the scheme $W$ is locally Cohen-Macaulay and equidimensional. The
scheme $W$ is a variety if and only if. Now we assume eo $<c’$ . Then $\overline{e_{0}}<t+1=N-m+1=$
N-dim((i ) ) $+1$ , which implies that the variety
in degenerate, namely there is a hyperplain$H$ including the variety. Then $V\subset(W_{0})_{red}$ implies that the variety $V$ is also degenerate, which
contradicts our assumption.
I$R_{V}$$S$$\mathrm{S}$$R_{V}$$\mathrm{F}_{V}$$V$$<-\cdot$$\mathrm{P}\mathrm{G}$$S$$R_{\mathcal{W}}$$R_{W}\mathrm{w}$$\mathrm{F}_{W}$$\oplus 5(-3)arrow-\cdot$$W_{0}$$V$$W_{0}$$\overline{e_{0},}=e’$$(W_{0})_{r\mathrm{e}d}$$0$$(W_{0})_{red}$Corollary 2.13 Let
be a variety of -genus zero and a scheme $W\subset P$ a pregeometricshell of V. Assume that $\dim(V)=\dim(W)$ . Then, $V=W.$$V\subset P=\mathrm{P}^{N}(\mathbb{C})$$\triangle$

Proof. By the assumption, using the claim (2.5.5), $\dim(R_{W})=$ dinl(RV) $=de_{J}pf_{J}h_{S_{+}}(R_{V})\leq depth_{S_{+}}(R_{W})$ ,which shows the ringis Cohen-Macauilay. Then, applying Proposition 2.12, we see that the scheme$W$ is a variety, which implies $V=$ $W$ .
I$R_{W}$To handle Hilbert polynomials efficiently, we prepare the following lemn .
$1\mathrm{a}$Lemma 2.14 (Finite factorial series expansion of Taylor type) Let us consider a polynomial ofreal coefficients $f(x)\in \mathbb{R}[x]$ of degree , in other words, a real valued function $f(x)$ defined on the fieldof real numbers (or on the ring of rational integers ) which has an expression by factorial monomials$(k=0,1, \ldots, r)$ :$r$$\mathbb{R}$$\mathbb{Z}$$x\mathrm{i}’ 1$$f(x)$

where the coefficient$+( \frac{c_{r-1}}{(r-1)!})x^{[r-1]}$$c_{0}+( \frac{c_{1}}{1!})x^{\lfloor 1]}\mathrm{T}$$=$$c_{r}A_{\mathrm{r}}(x)+c_{r-1}A_{r-1}(x)+\cdots+c_{1}A_{1}(x)+c_{0}$$c_{k}$$( \frac{c_{2}}{2!},)x^{\lfloor 2]}+\cdot$$k$$A_{k}(x)= \frac{x^{[k]}}{k!}=(\begin{array}{l}x+kk\end{array})$$c_{k}$$+( \frac{c_{r}}{r!})x^{1r\rfloor}\llcorner$,

is also a real number, and the -th factorial rnonomial$A_{k}.(x)$isof$1)(x+k-2)\cdot$ . $(x+1)$ and the Hilbert function$\cdot$

Then the coefficient

..

$=$

$x^{[k|}$means $(x+k)(x+k-$$\mathrm{P}^{k}(\mathbb{C})$$= \frac{(x+k)(x+k-1)\cdots(x+1)}{k!}$can be computed by using the (backward) difference operator,$c_{k}=(\nabla^{k}f)(-11$$\nabla$as follows.

138

holding if we
Proof. Analogously to the usual Taylor expansion, it is easy to see the formula ontake a notice on the facts that
Vl $=0,$ for a positive integer , $\nabla x^{[k]}.=k\cdot x[’-1]$ , namely$-k$ .
$\nabla A_{k}(x)=A_{k-1}(x)$ and that $A_{k}(a)=0$ if and only if $a=-1,$ -2, . . .
$c_{k}$$\nabla x^{\lfloor 0]}=$$k$$\mathrm{I}$$,$We should also recall a classical and well-known result for later use (cf. [8], [6]).
Lemma 2.15 (One point Chordal Variety) Let
be a non-degenerate closed subva$x_{0}\in V.$ then there is a projective line
sufficiently
we
take
a
point
generalsmoothof degree
Ifsatisfying the conditions: (a) this line contains the point
and another point $y\in V;(b)$ this line
itself is not contained in V. Now we put:
$V\subseteq P=\mathrm{P}^{N}(\mathbb{C})$$\geq 2.$$7ir^{\lrcorner}ty$

$L$

$L$

$x_{0}$$L$

Cd(xo, $V$ ) $:=$line(xo, ),$\cup$$x$$x\in V-\{x_{0}\}$namely the Zariski closure of the union of all the lines joining any point of $V-\{x_{0}\}$ and the point .
Then this variety Cd(xo, $V$ ) coincides with the projective cone with the vertexof the Zariski closure
theimage of $V-\{x_{0}\}$ projected from the center. and therefore $dim(Cd(x_{0}, V))$ $=dim(V)+$ 1.
ofMoreover, ate have: $deg(Cd(x_{0}, V))=deg(V)-$ . (In case of emphasizing this construction process, wewill tall this variety $Cd(x_{0}, V)$ as “one point chordal variety” with vertex
instead of “the cone” ).
$x_{0}$$x_{0}$$x_{0}$$1$$x_{0}$

We will use the next proposition including singular cases. For its proof, we refer to [10] and [8].
Proposition 2.16 Let be a complete intersection closed subscheme of type,... ,
inwith $dim(Y)=N-r\geq 3.$ Then $Pic(Y)\cong$ ZOy(1). Moreover, if the scheme
is a variety, then the
canonical bundle $K_{U}:=det(\Omega_{U}^{1})$ of the regular locus $U:=Reg(Y)$ of can be extended to the dualizing$(-N-1+5_{i=1}^{r}m_{i})$ of $Y$ .
lirle bundle$Y$$(m_{1}$$m_{r})$$P=\mathrm{P}^{N}(\mathbb{C})$

$Y$

$Y$

$O_{Y}$$\mathrm{i}$$K_{Y}^{\mathrm{o}}$Q3 Classification on$\mathrm{P}\mathrm{G}$-shells of.

$(\mathbb{P}^{1}, O_{\mathrm{P}^{1}}(4))$

Since a non-degenerate rational normal quartic curve
is uniquely determined up to theaction of $PGL(5, \mathbb{C})$ , we set up the circumstances as follows.
Let $X$ be a projective lineProjand embedded into the 4-th projective space:which has the tautological line bundle $O_{P}(1)=O_{P}(H)$ by using thearnple line bundle, or more precisely, by using the morphism in which the homogeneous
coordinates:::]. Then the homogeneous:
correspond to-Z2Z4,-Z2Z4,) ,-Z2Z4,
ideal
coincides with ( $-Z_{2}Z_{4},$
-ZOZ3
where
. Since the sheaf
has the CastelnuovO-Mumford regularity 1 with respect
the line bundle $O_{P}(H)$ , it is easy to see that the homogeneous coordinate ring $R_{X}=S/Ix$ has a minimal
graded -free resolution of 2-linear type:$X\subset P=\mathrm{P}^{4}(\mathbb{C})$$\mathrm{P}^{1}(\mathbb{C})=$$(\mathrm{C}[\mathrm{Z}0, T_{1}])$$P=\mathrm{P}^{4}(\mathbb{C})=Proj(\mathbb{C}[Z_{0}, \cdots, Z_{4}])$$O(4)=O_{\mathrm{P}^{1}(\mathbb{C})}(4)$$[Z_{0} :Z_{1} :

Z_{2} :$\mathrm{I}_{X}$

Z_{3} :

\mathrm{Z}\mathrm{A}]$$\lfloor\lceil T_{0}^{4}$$Z_{2}Z_{3}$$Z_{\mathrm{i}}^{2},$$S=\mathbb{C}[Z_{0}, \cdots, Z_{4}]$$T_{0}^{3}T_{1}$$Z_{1}Z_{3}$$T_{0}T\mathrm{j}$$T_{0}^{2}T_{1}^{2}$$T_{1}^{4}$$Z_{1}Z_{2}$$Z_{2}^{2}$$\mathrm{i}Z_{1}^{2}-Z_{0}Z_{2}$$O_{X}$$S$$\mathrm{F}\chi$.

where the maps$\varphi_{1}=\vee|\lceil Z_{1}^{2}-$$\mathrm{O}arrow$S) $i(i= 1,2, 3)$

$j$

ZYZ2,

Zl

$\underline{\varphi_{1}}\oplus^{6}S(-2)\underline{\varphi 2}\oplus^{8}S(-3)$$\underline{\varphi \mathrm{s}}\oplus^{3}0(4)arrow 0$,

are described by the matrices:$\mathrm{Z}2-\mathrm{Z}2$

Z3,$Z_{2}^{2}-Z_{0}Z_{4}$,$Z_{1}Z_{3}-Z_{0}Z_{4}$,$Z_{2}Z_{3}$-Z2Z4,$Z_{3}^{2}-Z_{2}Z_{4}$]$\cdot$$S$

139$\mathrm{j}$$2$$=[-Z_{0}-Z_{2}Z_{1}Z_{0}00$$-Z_{2}-Z_{0}Z_{1}000$$-Z_{3}-Z_{0}Z_{1}000$$-Z_{0}-Z_{3}Z_{2}000$$=[-Z_{0}-Z_{2}Z_{1}Z_{3}Z_{0}000$$\mathrm{p}_{3}$$-Z_{2}Z_{4}Z_{1}000$$-Z_{4}-Z_{2}-Z_{2}-Z_{0}Zz_{0}\mathrm{o}_{1}^{3}$$-Z_{3}Z_{2}Z_{4}000$$-Z_{3}Z_{1}Z_{4}000$$-Z_{3}-Z_{4}Z_{2}z_{4}^{0}\mathrm{o}]$$-Z_{2}-Z_{4}Z_{3}Z_{4}Z_{1}000]$.

Next we take ashell $W$ of the rational normal quartic curve $X\subseteq P$ and consider a minimal
graded -free resolutionof the homogeneous coordinate ringof $W$ . Then by the definition of-shell, we see easily that this complex (resolution) has a form:$\mathrm{P}\mathrm{G}$$S$$\mathrm{F}_{W}$$R_{W}$$\mathrm{P}\mathrm{G}$$\mathrm{F}_{W}$.

$0arrow$ S$\underline{\psi_{1}}$and is a subcomplex of the complex
follows.

aS{-2)$\mathrm{F}_{X}$.

$\underline{\psi_{2}}\oplus^{b}S(-3)\underline{\psi_{3}}\oplus^{c}S(-4)arrow 0,$. Then we can compute the Hilbert polynomial $A_{W}(x)$ of $W$ as

Lemma 3.1 Under the circumstances, using the integers $a\in\{0,1, \ldots 6\}$ , $b\in\{0,1, \ldots 8\}$ , $c\in\{0,1, \ldots 3\}$ .
the Hilbert polynomial $A_{W}(x)$ is written in the form:$A_{W}(x)=(1-a+b-c)A_{4}(x)+$ ( $2a-3b+$ 4c)Ai $(\mathrm{x})+(-a+3b-6c)A_{2}(x)+(-b+4c_{J})A_{1}(x)+(-c)$

Proof. From the projective sheafication of the minimal gradedhave the exact complex of sheaves:$S$-free resolution$0arrow \mathit{0}warrow \mathit{0}_{P}\underline{\psi_{1}}\mathrm{e}6\mathrm{o}\mathrm{P}(-3)\underline{\psi_{2}}\oplus^{b}O_{P}(-3)\underline{\psi_{3}}$-$\mathrm{F}_{W}$.

of the ring

$R_{W}$, we$\oplus^{c}O_{P}(-4)arrow 0,$

which shows that $A_{W}(x)=A_{4}(x)-a\cdot A_{4}(x-2)+b\cdot A_{4}(x-3)$ $-c\cdot A_{4}(x-4)$ . By applying Lemma 2.14,
we have the result.
$\iota$Now let us classify all the-shells of the rational normal quartic curve $X\subseteq P$ depending on
their codimensions. Here we should pay attention to the fact that the curve $X$ is a non-degeneratearithmetically normal non-singular projective variety of -genus zero and satisfies all the assumptions ofConjecture 0.1 except the last one (i.e. the existence of a shell frame ( $E$ , )).
$\mathrm{P}\mathrm{G}$$\triangle$.

$\sigma$

The Case of codim(W) $=1$

First we handle the case: codim(W) $=1.$ Before we apply Lemma 2.5 directly, we should take care thefact that for an arbitrary closed scheme
the condition that codim(W) $=1$ and the scheme $W$ is$V\subseteq P,$

140

a

-shcll of the scheme $V$ does not in general imply that the scheme $W$ is a divisor of $P$ since we do
not assume, for example, the scheme $W$ is equidimensional and so on. It may happen that the scheme$W$ has a primary componentof codimension 1 and has another component of codimension more than 1or an embedded component. Thus, to consider this case, we need the following lemma.
$\mathrm{P}\mathrm{G}$Lemma 3.2 Let
be a reduced and irreducible closed subscheme of, a closed subscheme
is of codimension 1 in the total space
and a pregeometric shell of the variety V. Then $W$ is anirreducible and reduced divisor of .
$V$

$P=\mathrm{P}^{N}(\mathbb{C})$$P$$\ddagger V$$P$

Proof. We have only to show that $dim_{(S/s_{+})}Tor_{1}^{S}$ (Rw, $S/S+$ ) $=1.$ Now assume that$.\mathrm{a}$$dim(s/s_{+})Tor_{1}^{6^{\mathrm{f}}}(R_{W}, S/S+)\geq 2.$

Then there are at least two equations {Gi,} of $W$ which is linearly independent in the $(S/S_{+})$ -vector$TorS1$$(R_{W}, S/S+)$ . By thespace
injectivity condition of-shells, we see that the equations
form a part of a minimal generators of the homogeneous idealof . Since the ideal
is a prime
ideal, both the equations
andare irreducible polynomials, otherwise they can not be a part ofminimal generators of the ideal . Since the equations : {Gi,} are linearly independent, they form
-regular sequence. Thus the closed subscheme $Y=\{G_{1}=G_{2}=0\}$ of is of pure codimension 2. On
the other hand, by its construction,and codim(Y) $=2,$ which is a contradiction.
1$G_{2}$$\mathrm{T}\mathrm{o}\mathrm{r}$$\{G_{1}, G_{2}\}$$\mathrm{P}\mathrm{G}$$V$$\mathrm{I}_{\mathrm{V}}$$G_{1}$$\mathrm{I}_{V}$$G_{2}$$G_{2}$$\mathrm{M}_{V}$

$S$

$P$

$W\subseteq Y$

Corollary 3.3 Let $W$ be a pregeometric shell of the rational quartic curveand of codimension
1 in P. Then the scheme $W$ is a reduced and irreducible quadric hypersurface of rank 3,4, or 5 whichmeans that $\triangle(W, O_{W}(1))$ $=0$ and the variety $W$ is arithmetically Cohen-Macaulay. Moreover, if thevariety $W$ be ashell of $X$ , then there is no shell frameof $X$ in $W$ .
$X\underline{\mathrm{c}}P$$G$$(E, \sigma)$

Proof. For the claim on the rank of quadric equations, we see that irreducibility of minimal generators
implies :rank
To see that every case occurs, we give examples of the quadric equations of therational quartic curve $X$ in Table 1.
$\geq 3.$rank
3
4
5(non-sing)quadric hypersurface$\supset X$$\mapsto Z_{1}-Z_{0}\overline{Z_{2}}$$Z_{1}Z_{3}-Z_{0}Z_{4}$$Z_{2}^{2}-$$2\overline{Z_{0}Z_{4}}1$$Z_{1}Z_{3}$Table 1: Examples of Quadrics
All the claims except the last one have been already proved in Lemma 3.2. Now we assume that there
exists a shell frameof $X$ in $W$ as in Definition 2.4. From the homomorphism :t $I_{X/W}\subseteq O_{W}$ ,we see that the norrnal bundle $N_{X/W}$ of $X$ in $W$ is isomorphic to. Putting $U:=Reg(W)$ and the
dualizing line bundle of $W$ to be(cf. Proposition 2.16), we have $O(-2)\cong K_{X}\cong K_{U}|_{X}\otimes detN_{X/W}\cong$((& $detE$, namely the line bundle $O(-2)$ can be extended to a line bundle& detE) (&on $W$ . By Proposition 2.16 again, we have an isomorphism $Pic\{W$ ). Then the line bundle
and therefore the line bundle $O(-2)$ can be extended to a multiple of the tautological line bundle $O_{P}(1)$of , which contradicts to the fact : $O_{X}(1)\cong O(4)$ .
1$(E, \sigma)$$\sigma$$E^{\vee}arrow$$E\otimes O_{X}$$K_{W}^{\mathrm{o}}$$K_{W}^{\mathrm{O}}$$\mathit{0}\chi$$L:=K_{W}^{\mathrm{o}}$$\cong \mathbb{Z}O_{W}(1)$

$P$

$L$

141

.

The Case of codim(W) $=3$

Next, skipping the most bothersome case of codim(W) $=2,$ we proceed to the case of codirn(W) $=3.$
Then we easily get the following result from Corollary 2.13.

Lemma 3.4 Let $W$ be a pregeometric shell of the rational quartic curve
P. Then the scheme $W$ coincides with the curve $X$ .

.

$X\subseteq P$

and of codimension 3 in

The Case of codim(W) $=2$

Now we has come to the remaining case: codim(W)$(a, b, c)$ in the order of handling in the sequel.

$=\underline{9}$. Let us list up all the cases of the triplet

Lemma 3.5 Let $W$ be a pregeometric shell of the rational quartic curveand of codimension 2,.
in P. Then all the cases of the $Tor$-Betti numbers $(a, b, c)$ of the minimal graded -free resolution
and of the Hilbert polynomials are listed in Table 2 below.
$X\subseteq P$$S$

Case No.

$(a, b, c)$

(1)
(2)
(3)

(4, 4, 1)(5, 6, 2)(3, 2, 0)$\mathrm{F}_{W}$$A_{W}(x)$$2A_{2}(x)-1$

A2 $(x)+2A_{1}(x)-$$2$$3A_{2}(x)-2A_{1}(x)$Table 2: Betti Numbers and Hilbert Polynomials
Proof. Since the degree of the Hilbert polynomial $A_{W}(x)$ is 2, applying Lemma 3.1, we have the equations below $((a, b, c)\in\{0,1, \ldots 6\}\cross\{0,1, \ldots 8\}. \cross \{0, 1, \ldots 3\} )$ , which are easily solved and bring Table
2.

$\{$

$1-a+b-c$$=$$2a-3b+4c$

$=$

$-a$

$4$

$3b-6c$

7

0
0
0
1

Now we will follow the order in Table 2 and handle each case, respectively.
so (1) The Case:(a, , ) $=(4,4.1)$
$b$

$c$

It is easy to see that this case never happens.
, we see that $deg(W)=2.$ On the other hand, we
Since the Hilbert polynomial $Aw(x)$ is A2
have $a=4,$ which means that there exit four -linearly independent quadric equations
generating the ho mogeneous ideal. Now, using the condition that $W$ is a-shell of the curve $X$ , werecall the proof of Lemma 3.2 and see that {Gi, } forms a -regular sequence and $Y=\{G_{1}=G_{2}=0\}$
is an arithmetically Cohen-Macaulay closed subscheme of pure codimension two. Obviously,$dim(W)=dim(Y)=2,$ and cleg(Yl) $=4.$ Now we take a primary componentof containing $X$ and$(\mathrm{x})-1$$\{G_{1}, G_{2}, G_{3}, G_{4}\}$$\mathbb{C}$$\mathrm{P}\mathrm{G}$$\mathrm{I}_{W}$$G_{2}$$S$$X\subset W\subseteq Y,$$Y_{0}$$Y$

142

considerafter putting the reduced structure on the space } . If dethen after
the process of Lomma 2.15, the one point chordal variety Cd(xo,) turns out to be a linear variety
of dimension 3. SinceCd(xo,), we see that the curve $X$ is degenerate, which
is a contradiction. Hence we have $de/$ ((YQ) )and the component
can not contain any main
component of $W$ . Now we have $W\cup$ (Yo)red
and therefore $deg(Y)\geq 5,$ which is absurd.
$(Y_{0})_{re}d$$\mathrm{o}$$|$$7((Y\mathrm{o})_{re\mathrm{Z}})$$|$$\leq 2,$$(Y_{0})_{r\mathrm{e}d}$$(Y_{0})_{red}$$X\underline{\subset}(Y_{0})_{red}\subseteq$$\geq 3$$\Gamma ed$..

$Y_{0}$$\subseteq Y$(2) The Case:(a, , ) $=(5,6.2)$
$b$

$c$

Using a rather delicate argument than the case above, we show that this case never occurs.

On the Hilbert polynomial, we know that $A_{W}(x)=A_{2}(x)+2A_{1}(x)$ $2=(1/2)x^{2}+(7/2)x+1,$ which
implies : $deg(W)=1.$ This shows us that the main component $F$ is only one and its structure sheaf as a
primary component of $W$ is already reduced and isomorphic to the projective plane. Since $X\subseteq W$
$X$
$F$
and
is non-degenerate, we have $Xl$
and $Y=X$ ’ $F\subseteq W,$ where the closed subscheme
is ascheme theoretic union of the closed subschemes $X$ and $F$ . At the generic point of the main componentof $W$ ,, which implies that the support of the ideal sheaf $I_{Y/W}$ does not contain the genericpoint , and therefore $dim(I_{Y/W})\leq 1.$ Then, the Hilbert polynomial of the ideal sheaf $I_{Y/W}$ is the form$:\chi(I_{YW}/(x))$ $=px+q$ (and $p\geq 0$ ) (cf. [11]). On the other hand, $XnF$ is finite number of points,
which implies that the Hilbert polynomial is awhere $k=lengthO_{X\cap F}$ . Now
lot us consider the following two exact sequences:-$\mathrm{P}^{2}(\mathbb{C})$$Y$$\langle$$\mathit{0}_{W,\zeta}\cong O_{Y,\zeta}$$\zeta$$p,$$r\in \mathbb{Z}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}:\chi(\mathrm{O}_{\mathrm{X}\cap \mathrm{F}}\urcorner(x))=k,$0 $arrow I_{Y/W}arrow \mathit{0}_{W}arrow \mathit{0}_{Y}arrow 0$$0arrow O_{Y}arrow O_{X}$ % $O_{F}arrow O_{X\cap F}arrow 0.$

Let us take their Hilbert polynomials, bind them up and get:
$A_{W}(x)$

))

{Of

$=$

$\mathrm{x}\{\mathrm{O}\mathrm{f}\{\mathrm{x}$$=$$\chi(O_{X}(x))+\chi(O_{F}(x))-\chi(O_{X\ulcorner 1F}(x))+\chi(I_{Y/W}(x))$$=\mathrm{x}$$\{\mathrm{x}))+\chi(I_{Y/W}(x))$$=$$A_{X}(m)+$ $4_{2}(x)-k+px$ $+q$$=$

$4x+1+$ (l/2) $(\mathrm{x}+2)(x+1)+px$$=$$(1/2)x^{2}+((11/2)+p)x+(2+q-k)$ .

$c$

$q-k$

Comparing the coefficient of the second term in this Hilbert polynomial with that of $Aw(x)$ previously
obtained, we see that (7/2)=((11/2)+p)\geq (ll/7), which is a contradiction.

..

(3) The Case:(a, , ) $=(3,2,0)$This case really occurs and is tllc most interesting from our view point.
Let us recall: $Ay/(x)=3\mathrm{A}2(\mathrm{x})-2A_{1}(x)$ , which implies $deg(W)=3.$ Moreover, the length of theminimal graded -free resolution
is 2, and therefore arith. $de,pth(W)=$ depth(R ) $=5-hds$ $(Rw)=$$5-2=3=dim(R_{W})$ . Thus the homogeneous coordinate ring $R_{W}$ is an arithmetically Cohen-Macaulay
ring. Applying Proposition 2.12, we see that the scheme $W$ is a variety of ( $W$, Aw(x)) $=0$ and of
$b$

$S$

$c$

,.

$\mathrm{F}_{W}$$y$$\triangle$

degree 3.
By the structure theorem on the projective varieties of -genus zero (cf. [6] or more classically [12]),the singular locus Sing(W) of $W$ is a linear space and the variety $W$ is the generalized projective coneover a non-singular projective variety $M$ (this can be obtained by a generic linear space section of $W$
which does not meet the linear space Sing(Wl ) of -genus zero with the vertex at their singular locus.
Since $dim(W)=2,$ we have only two cases : (3-1) $dim(Sing(W))=-1$ (namely $W$ is non-singular) andthe non-singular variety $M$ is $W$ itself; or (3-2) $dim(Sing(W))$ $=0$ (namely Sing(W) $=\{p_{0}\}$ ) and thenon-singular variety $M$ is a rational normal cubic curve.
$\triangle$$\triangle$

143

Moreover, the structure theorem says that in the case of (3-1) above, the polarized variety $(W, O_{W}(1))$, where the vector bundle $E$ is the one over a rational curve
is a rational scroll, the ample line bundle
and of the form:
is the relative tautologicalline bundle of the projective bundle
determined from the ample vector bundle $E$ . In te rnsof rational ruled surfaces, this variety $W$ is isomorphic to, a rational ruled surface of degree 1 and isembedded by a linear system 2 $f+C_{1}|$ .
On the other hand, in the case of (3-2), the blow-up :of the variety $W$ at the vertex
is obtained byand by a natural homomorphism. This variety $W$ is isomorphic to
on a rational curve, a rational ruled surface of degree 3
and the morphism :is given by a linear system $3f+C_{3}|$ .
Let us summarize these two cases in the following Table 3 and proceed to study how the curve $X$ isembedded in $W$ in each case.
$(\mathrm{P}(E), O_{\mathrm{P}(E)}(1))$$B=\mathrm{P}^{1}(\mathbb{C})$$E\cong O_{\mathrm{P}^{1}(\mathrm{C})}(2)\oplus O_{\mathrm{P}^{1}(\mathbb{C})}(1)$$o_{\mathrm{P}(E)}(1)$$\mathrm{P}(E)arrow B$$\Sigma_{1}$$\overline{W}arrow W\subseteq P$$q$$p_{0}$$\overline{W}\cong \mathrm{P}(O_{1\mathrm{I}^{\nu 1}(\mathbb{C})}(3)\oplus O_{\mathrm{P}^{1}(\mathrm{C})})$$\oplus^{5}\mathit{0}_{\mathrm{P}^{1}(\mathbb{C})}arrow O_{1\mathrm{P}^{1}(\mathrm{C})}(3)\oplus O_{1\mathrm{P}^{1}(\mathrm{C})}$$B=\mathrm{P}^{1}(\mathbb{C})$$q$$\Sigma_{3}$$\overline{W}arrow W\subseteq P$

Case No.

$dim(S^{\cdot}g(W))$

(3-1)
(3-2)

-1 (on-sing)0$\overline{W}$or$W\cong$$1\mathrm{i}1$$\mathrm{Y}$ear system (morph.)$|2f+C_{1}|$$\Sigma_{1}$(embedding)$|3f+C_{3}|$$\overline{W}\cong$$\mathrm{g}_{3}$Table 3: Cases of $(W, O_{W(}’1))$\ldotsThe Case (3-1)

Now we assume that the variety $W$ is isomorphic to the ruled surface
and embedded into by thelinear system $|2f+C_{1}|$ . Since $X\in|uf+vC_{1}|$ for some integer
the fact: $deg(X)=4$ and the Lemma
2.2 (2.2.3), we have : $(u, v)=(3,1)$ or $(2, 2)$ . Let us take a blow-down morphism :IIofcontracting the exceptional curve
to a point $p\in Y.$ Then the pull-back :of the tautologicalample line bundle of the projective plane $Y$ corresponds to the linear system $|f+C_{1}$ . Thus, if the curve$X\in|3f+C_{1}|$ , applying the projection formula to the curve $X$ with respect to the morphism , and thecomputation : $X.(f+C_{1})=3$ , $X.C_{1}=2$ show that the the curve $X$ comes from a singular irreducible
and reduced cubic plane curve passing through the point . Considering that 3 $f+C_{1}=(2f+C_{1})+f,$
namely a sum of the ample divisor 2 $f+C_{1}$ and an (effective) nef divisor , we see that the curve $X$ is a
nef divisor (an ample divisor).
By the similar argument, if the curve $X\in|2f+2C_{1}|$ , we see that the curve $X$ comes from a nonsingular conic which does not pass through the point . To see the curve $X$ is a nef divisor (N.B. notample e.g. $X.C_{1}=0$ ), we have only to apply the projection formula to a test curve with respect to themorphism since.
$P$$\Sigma_{1}$$u$$\mathrm{t}^{\mathrm{t}}$$b$$C_{1}$$arrow \mathrm{P}^{2}(\mathbb{C})=Y$$b^{*}O_{\mathrm{P}^{\underline{\circ}}(\mathbb{C})}(1)$

$|$

$b$

$p$

$f$

$p$

$b$

\ldots$X\in|/\mathrm{t}’ O_{\mathrm{P}}\circ\sim(\mathbb{C})(2)|$The Case (3-2)

Next we consider the case :theis isomorphic to theof the variety $W$ at the vertex
ruled surface
and the morphism : $Warrow W\subset P$ is given by the linear system $|3f+C_{3}|$ . Now,
taking the strict transform $q^{-1}(X)$ of the curve $X$ via the morphism , we seek the integers and suchthat $q^{-1}(X)\in|uf+vC_{3}|$ . Then the condition $deg(X)=4$ means $q^{-1}(X).(3f+C_{3})=4.$ Since thecurve $q^{-1}(X)$ is irreducible, we can apply Lemma 2.2 (2.2.3) to this case and get $(\iota x, \tau))=(4,1)$ . Then
X.C3 $=(4f+C_{3}).C_{3}=4-$ $=1,$ which shows the curve $X$ passes simply through the vertex .
$\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{w}\underline{- \mathrm{u}}\mathrm{p}\overline{W}$$p_{0}$$\Sigma_{3}$

$q$

$q$

$3$

4 Existence

$u$

$v$

$p_{0}$

Now let us show the existence of the three cases above and summarize the classification of the casec0dimiW)=2 we have already done.

144

Proj(S) and $Wa$
Proposition 3.6 Let $X$ be a non-degenerate quartic norrmal curve in
$X$
$W$
$=2.$
pregeometric shell ofwith codim(W)Then
is reduced and irreducible and a variety of A-genus
zero. The homogeneous coordinate ringof $W$ has a minimal graded -free resolution ,. with the$Tor$ -Betti numbers. There are three possible cases satisfying these properties as in Table 4.
$P=\mathrm{P}^{4}(\mathbb{C})=$$S$$R_{W}$$\mathrm{F}_{W}$$(3, 2_{\backslash }0)$

Case No.

$d.m(S^{\cdot}/(W))$(3-1-1)(3-1-2)(3-2)-1 (on-s. )-1 $(0-\cdot. g)$$W$

or $W$$\cong\Sigma_{1}$$g$$W\cong\Sigma_{1}$

–0

$l$

.

ear $sytem(mo h.)$$|\mathit{2}f+C\mathrm{l}$$|$(embeddi$|2f+C\mathrm{l}$ $|(emb\overline{edd}.

$g$

)

g)$

or

llnear system(ample)$X\in|2f+2C_{1}|$ ($nef$, not ample)$X$$q^{\overline{-1}}(X)\in|4f+C_{3}|$$|3f+C_{3}|$$–\cong\Sigma$$q^{-\mathrm{I}}(X)\in-$$X\in|37$ $+C_{1}|$Table 4: Cases of a surface W and a curve $X$
Conversely, there exist a non-singular projective surface $W$ and a non-singular projective curve $X$ on $W$
which satisfy the conditions in Table 4 above. Moreover, once if the surface $W$ and the curve $X$ on $W$
are given as in Table 4 above, then the surface $W$ is always a pregeometric shell of the curve $X$ (cf. also

\S 4 ).
$\cdot$Proof. It is enough to show the existence part of the claim above. If we do not assume the condition of“pregeometric shells”,, Lemma 2.2 (2.2.3) shows the existences of pairs of a non-singular projective curveand a non-singular projective surface: $X\subset W$ in
as in Table 4. Then, the curve $X$ is a curveof degree 4 and the surface $W$ is of degree 3. By this construction, the surface $W$ is a linearly normal, i.e.
a natural map $H^{0}(P, O_{\mathrm{P}}(1))arrow H^{0}(W, O_{W}(1))$ is surjective. On the non-degeneracy of $X$ , using Table
4, we can check it easily by computing $H^{0}(W, I_{X/W}(H))\cong H^{0}(W, O_{W}(-X+H))$ $=0$ (if) or$H^{0}(\overline{7\overline{V}}, I_{q}1(X)/\tilde{W}(q^{*}H))\cong H^{0}(\overline{W}, O_{\overline{W}}(-q^{-1}(X)+q^{*}H))=0$
E3). This also shows the linear(if
normality of the rational curve $X$ (adjunction formula and Lemma 2.2 (2.2.4)).
Let us show that the surface $W$ is a pregeometric shell of $X$ in these three cases. In every case, boththe surface $W$ and the curve $X$ are varieties of -genera zero. Now we refer to the book [6] and apply
its results: (4.12) Corollary and the argument of (5.1), which imply that both varieties $X$ and $W$ arewith metically Cohen-Macaulay. Then the result of [5] shows that the homogeneous coordinate rings
has 2-lincar minimal graded -free resolution. Applying Proposition 2.5 (2.5.8), we see that the surface
It is a pregeometric shell of $X$ .
$P=\mathrm{P}^{4}(\mathbb{C})$$W\cong\Sigma_{1}$$\overline{W}\cong$$\triangle$$R_{W}$$S$$\mathrm{I}$The argu nent in the proof of Proposition 3.6 brings also the following useful result.

Corollary 3.7 Let $V\subset W$ be closed subschemes ofis non. Assume that the scheme
deqcneratc and the scherne $W$ is linearly normal and of a variety of -genus zero. Then the variety $W$
is a pregeometric shell of .
$V$

$P=\mathrm{P}^{N}(\mathbb{C})$$\triangle$$V$

Corollary 3.8 Let
be a non-degenerate, linearly no rmal closed subvariety of codimension and of -genus zero. Then, there is a chain of varieties:$V\subset P=\mathrm{P}^{N}(\mathbb{C})$$r$$\triangle$$V=W_{0}\subset W_{1}\subset W_{2}\subset\cdots\subset W_{k}\subset\cdots\subset W_{r}=P,$

where the variety$W_{k}$$(k=0,1, \cdots, r)$ is a pregeometric shell

of

$V$

with $cod_{i}m(W_{k}, P)$ $=r-k.$

Proof. Starting from the variety $V$ , we first construct inductively a chain of varieties-genus zero. As an induction hypothesis, we assume that we have a chain of varieties :$\triangle$$\{W_{k}\}_{k=0}^{r}$

with

145

$V=W_{0}\subset W_{1}\subset\cdots\subset W_{k_{0}}$,$(k= 0, 1, \cdots, k_{0})$ are varieties ofwhere all the varieties-genus zero. Then we take a nonsingular.
point of the variety . and set
to be a one point chordal variety
with the vertex. Then it is easy to see that the variety
is also of A-genus zero. Thus we obtain a chain of thevarieties
with -genus zero. Then we apply COrOllary3.7 to each pair $V\subset W_{k}$ and get what
we wanted.
$W_{k}$$x$$\triangle$$W_{k_{0}+1}$$W_{k_{0}}$$x$$Cd(x, \mathrm{W}\mathrm{k}\mathrm{O})$$W_{k_{\mathrm{O}}+1}$$\triangle$$\{W_{k}\}_{k=0}^{r}$

Q4 Explicit Examples.
Ill this section, using homogeneous coordinates explicitly, we give an example of a pregeometric shell ofcodimension two for the rational quartic curve $X$ , which appeared as one of the three cases in the list of
PrOpOsitiOn3.6. On explicit examples of the rest two cases, see [23]. The equations for $X$ and so on are
the same as in the first part of the previous section.

Example 4.1 (The Case:(3-1-1)) Taking,), we get a surface $W$ as
Proj(Cl
the blow-up of $Y$ at the center $p_{1}=[1$ : 0:0 . The embedding of $W$ by the linear system $|2f+C_{1}|$ , namely
the linear system coming from the conies passing through the point , is given by [::.
:$Z_{4}]=$ [UOU2 : UOU2 :::]. The equations of $W$ are : $Z_{1}Z_{2}-Z_{0}Z_{3}=0$ , $Z_{1}Z_{3}-Z_{0}Z_{4}=0.$Z2Z4 $=0.$ Then a minimal graded -free resolution of the homogeneous coordinate ringof $W$:$Y=\mathrm{P}^{2}(\mathbb{C})=$$[U_{0},$$U_{1}$$U_{2}]$$]$$Z_{0}$$p_{1}$$U_{1}^{2}$$U_{1}U_{2}$$Z_{1}$$Z_{\mathit{2}}$$Z_{3}$$U_{2}^{2}$$S$$R_{\mathcal{W}}$$Z_{3}^{2}-iS$$\mathrm{F}_{\mathfrak{l}V}$

where the maps$\psi_{i}$.

O $arrow$ S-$\oplus 2S(-3)$$arrow 0,$$Z_{3}^{2}-Z_{2}Z_{4}$]$\varphi$$\underline{\Psi 1}\oplus^{3}S(-2)$(i $=1,$ 2) are described by the matrices:

t&l

$=$

[

$Z_{1}Z_{2}$–ZOZ3$Z_{1}Z_{3}$–ZOZ4$/\mathrm{j})_{2}=\{\begin{array}{ll}-Z_{3} Z_{4}Z_{2} -Z_{3}-Z_{0} Z_{1}\end{array}\}$The curve $X$ comes from a curve $[U_{0} : U_{1} : U_{2}]=[T_{0}^{3} : T_{0}T_{1}^{2} : T_{1}^{3}]iY$ , namely a singular cubicwith a cusp at the point $p_{1}=[1$ : 0:0 .

$U_{1}^{3}-U_{0}U_{2}^{2}=0$$]$

The $Tor$ injectivity condition can be confirmed by using a homomorphism of complexes:$\mathrm{F}_{X}$.
.

$\mathrm{F}_{W}$$\mathrm{O}arrow S\underline{\varphi_{1}}\oplus^{6}S(-2)\underline{\varphi_{2}}\oplus^{8}S(-3)\underline{\varphi_{3}}\oplus^{3}S(-4)arrow 0$$||$$0arrow$ S$\uparrow\nu_{1}$$\uparrow U2$$\underline{\psi_{1}}\oplus^{3}S(-2)\underline{\psi_{2}}\oplus^{2}\mathrm{S}(-3)$

where the matrices
rnatnces for the maps $\nu_{i}(i=1,2)$ are:$arrow$0,

48$\nu_{1}=$$\{\begin{array}{lll}0 0 01 0 00 0 00 1 00 0 00 0 1\end{array}\}$$\nu_{2}=|00000001$$00000001$

References[1] T. Ashikaga : private correspondence on “singular ambient filtrations for a rational normal curve”(will be published in somewhere).
[2] W. Bruns and J. Herzog : Cohen-Macaulay Rings, Cambridge studies in advanced mathematics 39,
Cambridge University Press, (1993).
[3] D. Eisenbud : Green’s conjecture, in “Free Resolutions in Commutative Algebra and Algebraic
Geometry”, (Editors) D. Eisenbud et al. Boston, MA (1992).

[4] D. Eisenbud and S. Goto :Linear
pp.
: Linear free resolutions and minimal multiplicity, J. of Algebra 88, pp
89-133 (1984).
[5] D. Eisenbud, 0. Riemenschneider and F. 0. Schreyer :Resolutions: Resolutions of Cohen-Macaulay algebras,
Math.Ann.257, pp. 85-98 (1981).
[6] T. Fujita :Classification: Classification theories of polarized varieties, London Math. Soc. Lect. Note Ser. 155,
Cambridge Univ. Press (1990).

[7] M. Green:Koszul
Green : Koszul cohomology and geometry of I)rojective
projective varieties, J. Differ. Geom. 19, pp. 125-171(1984).
[8] R. Hartshorne :Algebraic: Algebraic Geometry, GTM52, Springer-Verlag, (1977).

[9] H. Hironaka :Anote:A note on algebraic geometry over ground rings, The invariance of Hilbert characteristic functions under the specialization process, 111. J. Math. 2, pp.355-366, (1958).
[10] R. Hartshorne: Ample subvarieties of algebraic varieties, Springer Lecture Notes 156, SpringerVerlag, (1970).
(1966).
[11] S. L. Kleiman :Toward: Toward a numerical theory of ampleness, Ann. of Math., 84 pp. 293-344 (1966)[12] M. Nagata: On rational surfaces I, II, hIenl.
Mem. Coll. Sci. Kyoto (A) 32 pp. 351-370 (1960), and 33
pp.271-293 (1960).

[13] K. Petri:
Petri : Uber die invariante Darstellung algebraischer Funktionen einer Variablen, Math. Ann. 88,
pp. 243-289 (1923).
[14] Z. Ran :Absence: Absence of the Veronese from Smooth Threefolds in(1991).

$\mathrm{P}^{5}$, Contemp. Math. 116, pp.125-128[15] B. Saint-Donat :On: On Petri’s analysis of the linear system of quadrics through a canonical curve,
Math. Ann. 206 pp. 157-175 (1973).

147[16] F. O. Schreyer : Syzygies of canonical curves and special linear series, Math. Ann. 275, pp. 105-137(1986).
[17] J.Stiickrad and W. Vogel : Castelnuovo bounds for locally Cohen-Macaulay schemes. Math. Nachr.
136, pp. 307-320 (1988).
[18] T. Usa : Obstructions of infinitesimal lifting, Comm. Algebra, 17(10), pp. 2469-2519 (1989).
[19] T. Usa : An example of filtrations on syzygies induced by meta-Lefschetz operators, Report Sci. H.
I. T., N0.8, pp. 12-25 (1997).
[20] T. Usa : Problems on geometric structures of projective embeddings, Report Sci. H. I. T. . N0.9,
pp. 12-29 (1998) (Improved version : Duke -print).
$\mathrm{E}$$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{A}\mathrm{G}/0001004$[21] T. Usa :2-bundles constructed from projected canonical curves in
No. 11, pp. 1-10 (2000).

$\mathrm{P}^{3}(\mathbb{C})$, Report Sci. H. I. T. .

[22] T. Usa : Infinitesimal Syzygy Bundles, Report Sci. H. I. T. . N0.12, pp. 1-17 (2001).
[23] T. Usa : A Classification of Pregeometric Shells of a Rational Quartic Curve, Report Sci. H. I. T. .
N0.13, pp. 1-17 (2002).

[24] T. Usa : A Classification of Pregeometric Shells of a Veronese surface (in preparation).