数理解析研究所講究録
第 1689 巻 2010 年 16-25

16

Lefschetz elements of Artinian Gorenstein
algebras and Hessians of homogeneouspolynomials
Toshiaki Maeno
Department of Electrical Engineering
Kyoto University, Kyoto 606-8501, Japan

This article is based on my joint work with Junzo Watanabe [8]. The Lefschetz property is a ring-theoretic abstraction of the Hard Lefschetz Theorem
for compact K\"ahler manifolds. The following are fundamental problems on
the study of the Lefschetz property for Artinian graded algebras:

Problem 0.1. For a given graded Artinian algebra
has the strong (or weak) Lefschetz property.

$A$, decide whether or not$A$

Problem 0.2. When a graded Artinian algebra has the strong Lefschetzof degree
property, determine the set of Lefschetz elements in the part
one.
$A$

$A_{1}$In this work, we give a characterization of the Lefschetz elements in Artinian Gorenstein rings over a field of characteristic zero in terms of thehigher Hessians. As an application, we give new examples of Artinian Gorenstein rings which do not have the strong Lefschetz property.
$k$1

Lefschetz properties$A_{D}\neq 0$ , be a graded Artinian algebra.
Definition 1.1. Let(1) We say that has the strong Lefschetz property if there exits an element$L\in A_{1}$ such that the multiplication map$A=\oplus_{d=0}^{D}A_{d},$$A$$\cross L^{d}:A_{i}arrow A_{i+d}$17
is of full rank (i.e. injective or surjective) for all $0\leq i\leq D$ and $0\leq d\leq D-i$ .
We call $L\in A_{1}$ with this property a strong Lefschetz element.
(2) If we assume the existence of $L\in A_{1}$ such that$\cross L:A_{d}arrow A_{d+1}$

is of full rank for $d=0,$
property.

$\ldots,$$D-1$ , we say that$A$

has the weak Lefschetz

In the following sections, we mainly investigate the strong Lefschetz property for Artinian Gorenstein algebras.

2

Artinian Gorenstein algebra

Definition 2.1. (See [10, Chapter 5, 6.5].) A finite-dimensional graded kalgebra $A=\oplus_{d=0}^{D}A_{d}$ is called the Poincar\’e duality algebra if $\dim_{k}A_{D}=1$
and the bilinear pairing$A_{d}\cross A_{D-d}arrow A_{D}\cong k$

is non-degenerate for $d=0,$$\ldots,$$[D/2]$ .

We will need two characterizations of the Artinian Gorenstein algebrain order to show our main theorem in the next section. The proofs of thefollowing propositions also can be found in [8].
Proposition 2.1. (See [3].) A graded Artinian k-algebra
duality algebra if and only if is Gorenstein.

$A$

is a Poincar\’e$A$

Proposition 2.2. (See [1],[2],[4].) Let I be an ideal of $Q=k[X_{1}, \ldots, X_{n}]$and $A=Q/I$ the quotient algebra. Denote by $m$ the maximal ideal $(X_{1}, \ldots, X_{n})$is Gorenstein if and only if there
and the k-algebraof Q. Thenexists a polynomial $F\in R=k[x_{1}, \ldots, x_{n}]$ such that $I=Ann_{Q}F$.
$\sqrt{I}=\mathfrak{m}$3$A$Characterization of Lefschetz elements

In this section we discuss the set of the Lefschetz elements for graded Artinian
Gorenstein rings $A=k[X_{1}, \ldots, X_{n}]/Ann_{Q}F$ with the standard grading.

18
Definition 3.1. Let $G$ be a polynomial in $k[x_{1}, \ldots, x_{n}]$ . When a familyof homogeneous polynomials of degree $d>0$ is given, we callthe polynomial$B_{d}=\{\alpha_{i}^{(d)}\}_{i}$$\det((\alpha_{i}^{(d)}(X)\alpha_{j}^{(d)}(X)G(x))_{i,j=1}^{\#B_{d}})\in k[x_{1}, \ldots, x_{n}]$. We, and denote it by
the d-th Hessian of $G$ with respect tois clear.
denote the d-th Hessian simply by $Hess^{(d)}G$ if the choice of$Hess_{B_{d}}^{(d)}G$$B_{d}$$B_{d}$

Theorem 3.1. ([11, Theorem 4], [8, Theorem 3.1]) Fix an arbitrary k-linear$[D/2]$ . An element $L=a_{1}X_{1}+\cdots+a_{n}X_{n}\in A_{1}$basisof for $d=1,$
is a strong Lefschetz element of $A=Q/Ann_{Q}F$ if and only if $F(a_{1}, \ldots, a_{n})\neq$
and$B_{d}$$A_{d}$$\ldots,$$0$$(Hess_{B_{d}}^{(d)}F)(a_{1}, \ldots, a_{n})\neq 0$

for $d=1,$$\ldots,$$[D/2]$ .

The proof is easy if we take the propositions in the previous section for
by $[\omega(X)]$ $:=\omega(X)F(x)$ for:
granted. Fix the identification
any $\omega(X)\in A_{D}$ . Note that $\omega(X)F(x)\in k$ , because $\deg\omega=\deg F=D$ .
Since is a Poincar\’e duality algebra, the necessary and sufficient condition
for $L=a_{1}X_{1}+\cdots+a_{n}X_{n}\in A_{1}$ to be a strong Lefschetz element is that thebilinear pairing
$[$

$]$

$A_{D}arrow\sim k$$A$$A_{d}\cross A_{d}$$arrow$$A_{D}$$(\xi, \eta)$$\mapsto$$L^{D-2d}\xi\eta$

is non-degenerate for $d=0,$
and only if the matrix$\ldots,$$\cong$$k$$\mapsto$$[L^{D-2d}\xi\eta]$$[D/2]$ . Therefore $L$ is a Lefschetz element if$(L^{D-2d}\alpha_{i}^{(d)}(X)\alpha_{j}^{(d)}(X)F(x))_{ij}$has nonzero determinant. For a homogeneous polynomial $G(x_{1}, \ldots, x_{n})\in$$k[x_{1}, \ldots, x_{n}]$ of degree, we have the formula$d$$(a_{1}X_{1}+\cdots+a_{n}X_{n})^{d}G(x_{1}, \ldots, x_{n})=d!G(a_{1}, \ldots, a_{n})$,so$L^{D-2d}\alpha_{i}^{(d)}(X)\alpha_{j}^{(d)}(X)F(x)=(D-2d)!\alpha_{i}^{(d)}(X)\alpha_{j}^{(d)}(X)F(x)|_{(x_{1},\ldots,x_{n})=(a_{1},\ldots,a_{n})}$This completes the proof of the theorem.

.

19

4

Set of Lefschetz elements

In this section we discuss the set of Lefschetz elements for some simple examples of Gorenstein algebras with the strong Lefschetz property based on
Corollary 3.1.

Example 4.1. Let us consider the Gorenstein ring $A=k[X_{1}, \ldots, X_{n}]/Ann_{Q}F$
associated to the Fermat type polynomial$F= \sum_{i=1}^{n}x_{i}^{n}-n(n-1)s\prod_{i=1}^{n}x_{i}$,where $s\in k$ is a parameter. One can check that has the strong Lefschetz
property for any $s\in k$ by computation of Hessians. Let us see the explicit
condition for the Lefschetz element for $n=3$ . For the polynomial $F=$$x^{3}+y^{3}+z^{3}-6s\cdot xyz,$
has the following structure:
$A$

$A$

Case $s^{3}\neq 0,1$ ,Case $s=0$ ,Case $s^{3}=1$ ,$A\cong k[X, Y, Z](sX^{2}+YZ, sY^{2}+XZ, sZ^{2}+XY)$ ,$A\cong k[X, Y, Z](X^{3}-Y^{3}, X^{3}-Z^{3}, XY, YZ, XZ)$ ,$A\cong k[X, Y, Z]’(sX^{2}+YZ, sY^{2}+XZ, sZ^{2}+XY, XZ^{2}, YZ^{2})$ .

The Hilbert function of is Hilb $(A)=(1,3,3,1)$ for all $s\in k$ . The condition
for $L=aX+bY+cZ\in A_{1}$ to be a strong Lefschetz element is that$A$$a^{3}+b^{3}+c^{3}-6s\cdot abc\neq 0$

and$s^{2}a^{3}+s^{2}b^{3}+s^{2}c^{3}-(1-2s^{3})abc\neq 0$.

This means that the projectivization of the set of non-Lefschetz elements inis the union of two elliptic curves intersectiong at each other’s
inflection points.

$\mathbb{P}(A_{1})\cong \mathbb{P}^{2}$

Example 4.2. In [7], the set of the Lefschetz elements for the coinvariant. Let bealgebra of the finite Coxeter group is determined except for typethe standard reflection representation of the finite irreducible Coxeter group$W$. Then $W$ acts on the polynomial ringand the W-invariant,
subalgebra
is generated by the fundamental W-invariants$r=\dim V$. The coinvariant algebra
is defined as the quotient algebra$H_{4}$$V$$R=Sym_{\mathbb{R}}V^{*}$$R^{W}$$f_{1},$$R_{W}$$\ldots,$$f_{r}$

20. It is known that $R_{W}$ is Gorenstein (see e.g.

[10, Theorem
is isomorphic to the cohomology
is crystallographic,
7.5.1] . Whenring of the corresponding flag variety. In [7], it was shown that the set ofLefschetz elements in $V^{*}=(R_{W})_{1}$ is the complement of the union of thereflection hyperplanes. For crystallographic case, their argument is basedon the ampleness criterion for the -divisors on the flag variety, so it isapplicable only when the field of coefficients is the field of real numbers.
Let us consider the case $W=S_{3}$ and$R/(f_{1}, \ldots, f_{r})$

$W$

$)$

$R_{W}$$\mathbb{R}$$k$$\mathbb{R}$$R_{W}=\mathbb{R}[X, Y, Z]/(X+Y+Z, XY+YZ+ZX, XYZ)$ .

with $\triangle=(x-$is also given by $R_{W}=\mathbb{R}[X, Y, Z]/$ Ann
has a linear basis $B_{1}=\{X, Y\}$ .
$y)(x-z)(y-z)$ . The degree one part
Then we have

The algebra$\triangle$$R_{W}$$(R_{W})_{1}$$Hess_{B_{1}}^{(1)}\triangle=-4(x^{2}+y^{2}+z^{2}-xy-yz-zx)$ ,which is a negative definite quadratic form. Hence the set of the Lefschetz
elements is given by$\{(x, y, z)|\triangle(x, y, z)\neq 0\}\subset V^{*}$.

, we have to take care of the condition $x^{2}+y^{2}+z^{2}-xy-$
If we work in, too.
yz–zx$V_{\mathbb{C}}^{*}$$\neq 0$Remark 4.1. Recently, a purely algebraic proof of the strong Lefschetz
property for coinvariant algebras of finite Coxeter groups has been given
by McDaniel [6] except for types$E_{8}$

and$H_{4}$.

associated to a fan
Example 4.3. Let us consider the toric variety
is compact and nonsingular.
. We assume thatof the latticebe the set of l-dimensional cones (rays) of . EachLet$\rho\in\Sigma(1)$. Denote byray
determines a torus invariant divisior $D(\rho)$ on$n(\rho)\in N$ the primitive generator of a ray. Let $Q$ be a polynomial. We defines the ideals $I,$ $J\subset Q$ by
ring in the variables$\Sigma$$X_{\Sigma}$$N=\mathbb{Z}^{r}$$X_{\Sigma}$$\Sigma$$\Sigma(1)=\{\rho_{1}, \ldots, \rho_{k}\}$$X_{\Sigma}$$\rho\in\Sigma(1)$$X_{\rho},$$\rho\in\Sigma(1)$$I:=( \sum_{\rho\in\Sigma(1)}\langle m, n(\rho)\rangle X_{\rho}|m\in M)$

,

and
$J$

$:=$

(

$X_{\rho_{1}}\cdots X_{\rho_{s}}|\rho_{1},$$\ldots,$$\rho_{s}$are distinct and do not generate a cone in$\Sigma$).

21
Then the cohomology ring of$X_{\Sigma}$has the following structure:.

$H^{*}(X_{\Sigma})\cong Q/I+J$

If we define a polynomial $F(x)$ as the intersection number,$F(x):=( \sum_{i=1}^{k}x_{k}D(\rho_{i}))^{r}$then we have a presentation of the cohomology ring of$X_{\Sigma}$

as follows:.

$H^{*}(X_{\Sigma})=Q/Ann_{Q}F$

I. (Hirzebruch surfece)
For an integer , consider the complete fan of
the rays$a$$N=\mathbb{Z}n_{1}+\mathbb{Z}n_{2}$$\rho_{1}:=\mathbb{R}_{+}n_{1},$ $\rho_{2}:=\mathbb{R}_{+}n_{2},$ $\rho_{3}:=\mathbb{R}_{+}(-n_{1}+an_{2}),$determined by$\rho_{4}:=\mathbb{R}_{+}(-n_{2})$The corresponding toric surface is the Hirzebruch surface
structure of a -bundle over :$\mathcal{F}_{a}$.

, which has the$\mathbb{P}^{1}$$\mathbb{P}^{1}$$\mathcal{F}_{a}=\mathbb{P}(\mathcal{O}\oplus \mathcal{O}(a))arrow \mathbb{P}^{1}$.

$:=$is generated by the classes of the fiber
The cohomology ringand the section $s:=D(\rho_{2})$ . We can see that the intersection form on
is given by$f$$H^{*}(\mathcal{F}_{a})$$D(\rho_{1})$$H^{2}(\mathcal{F}_{a})$$f^{2}=0,$ $fs=1,$ $s^{2}=-a$.

Then it is easy to see the following:
and $\beta>0$ .
is ample(i) The element.
andis Lefschetz(ii) The element
Though the classes of form $\alpha(f+as),$ $\alpha>0$ , belong to the boundary of theample cone, they are Lefschetz. In this example, we observe that points on
the wall of the ample cone may be the Lefschetz element.
$\alpha f+\beta s\in H^{2}(\mathcal{F}_{a})$$\Leftrightarrow\alpha>a\beta$$\alpha f+\beta s\in H^{2}(\mathcal{F}_{a})$$\Leftrightarrow\beta\neq 0$$2\alpha\neq a\beta$

II. (non-projective toric 3-fold.)
The simplest example of non-projective compact toric manifolds is due tobe the lattice of rank 3. We
Miyake and Oda [9]. Let$N=\mathbb{Z}n_{1}+\mathbb{Z}n_{2}+\mathbb{Z}n_{3}$22define a complete fan
and their faces:$\Sigma$of $N$ consisting of the following 3-dimensional cones$\mathbb{R}_{+}n_{1}+\mathbb{R}_{+}n_{2}+\mathbb{R}_{+}n_{3}$,$\mathbb{R}_{+}n_{1}+\mathbb{R}_{+}n_{2}+\mathbb{R}_{+}n_{1}$,$\mathbb{R}_{+}n_{2}+\mathbb{R}_{+}n_{3}+\mathbb{R}_{+}n_{2}$$\mathbb{R}_{+}n_{1}+\mathbb{R}_{+}n_{3}+\mathbb{R}_{+}n_{3}$,$\mathbb{R}_{+}n_{1}+\mathbb{R}_{+}n_{1}+\mathbb{R}_{+}n_{3}$,$\mathbb{R}_{+}n_{2}+\mathbb{R}_{+}n_{2}+\mathbb{R}_{+}n_{1}$$\mathbb{R}_{+}n_{3}+\mathbb{R}_{+}n_{2}’+\mathbb{R}_{+}n_{3}$,$\mathbb{R}_{+}n_{0}+\mathbb{R}_{+}n_{1}’+\mathbb{R}_{+}n_{2}’,$$\mathbb{R}_{+}n_{0}+\mathbb{R}_{+}n_{2}’+\mathbb{R}_{+}n_{3}$$\mathbb{R}_{+}n_{0}+\mathbb{R}_{+}n_{1}’+\mathbb{R}_{+}n_{3}$

,

,

,
,

where
$n_{0}:=-n_{1}-n_{2}-n_{3},$ $n_{1}:=n_{0}+n_{1},$ $n_{2}’:=n_{0}+n_{2},$ $n_{3}:=n_{0}+n_{3}$ .

There exist no strictly upper convex -linear support functions, so the fan. Let us denote
gives an example of compact non-projective toric manifold
hasby$i=0,1,2,3$ , the rays of . The cohomology ring of
the presentation$\Sigma$$\Sigma$$X_{\Sigma}$$\rho_{i}$$X_{\Sigma}$$\Sigma$$:=\mathbb{R}_{+}n_{i},$,$H^{*}(X_{\Sigma})=\mathbb{C}[X_{0}, X_{1}, X_{2}, X_{3}]/AnnF$

where$F=(x_{0}D(\rho_{0})+x_{1}D(\rho_{1})+x_{2}D(\rho_{2})+x_{3}D(\rho_{3}))^{3}$$=-4x_{0}^{3}-x_{1}^{3}-x_{2}^{3}-x_{3}^{3}+3x_{1}x_{2}x_{3}$.

Since the Hessian of $F$ is given by$Hess(F)=-1296x_{0}(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}-3x_{1}x_{2}x_{3})$,ishas the strong Lefschetz property, thoughconsists of
non-projective. The set of non-Lefschetz elements infour hyperplanes and one cubic hypersurface.

we can see that

$X_{\Sigma}$$H^{*}(X_{\Sigma})$$H^{2}(X_{\Sigma}, \mathbb{C})$5

Gorenstein algebras which do not have thestrong Lefschetz property

The result in Section 3 shows that a polynomial $F$ gives an example ofGorenstein algebra which does not have the strong Lefschetz property if
one of the higher Hessians of $F$ is identically zero. The polynomial $F=$$x_{0}u^{2}+x_{1}uv+x_{2}v^{2}$ is the simplest example whose Hessian vanishes, but no23
variables can be eliminated by a linear transformation of the variables (see[11, Example 1] .
Here we give examples of forms $F$ such that $HessF\neq 0$ and $Hess^{(2)}F=$. By using these forms we can also give examples of Gorenstein algebras$A=Q/Ann_{Q}F$ which do not satisfy the strong Lefschetz property.
$)$

$0$

Example 5.1. Let us consider the polynomial$F$ $:= \sum_{j=0}^{n}x_{j}^{2}u^{n-j}v^{j}\in k[u, v, x_{0}, \ldots, x_{n}]$and the corresponding algebra $A=Q/Ann_{Q}F$, where $Q=k[U, V, X_{0}, \ldots, X_{n}]$ ,. The Hessian of $F$ with respect to theandbasis $U,$of degree one is expressed as follows:$U=\partial/\partial u,$$V=\partial/\partial v$$V,$ $X_{0},$$\ldots,$$X_{i}=\partial/\partial x_{i}$$X_{n}$$HessF=$$2^{n+1}(uv) \frac{n(n-1)}{2}\{(\sum_{j=0}^{n-1}(n-j)(n-j+1)x_{j}^{2}u^{n-j-1}v^{j})(\sum_{j=1}^{n}j(j+1)x_{j}^{2}u^{n-j}v^{j-1})$$-uv( \sum_{j=1}^{n-1}j(n-j)x_{j}^{2}u^{n-j-1}v^{j-1})^{2}\}\neq 0$.

On the other hand, we can see that the second Hessian is identically zero,i.e. $Hess^{(2)}F=0$ in $k[u, v, x_{0}, \ldots, x_{n}]$ . This means that the algebra
does
not have the strong Lefschetz property.
$A$

Example 5.2. There exists an example of a polynomial $F$ of degree 5 with
5 variables such that $HessF\neq 0$ and $Hess^{(2)}F=0$ . Let us choose$F=x^{2}u^{3}+xyu^{2}v+y^{2}uv^{2}+z^{2}v^{3}\in k[u, v, x, y, z]$.

Then$U,$ $V,$ $X,$ $Y,$

$Z\in A=k[U, V, X, Y, Z]/Ann_{Q}F$

are linearly independent. So we have
$HessF=48u^{3}v^{3}(u^{5}x^{4}+8u^{4}vx^{3}y+16u^{3}v^{2}x^{2}y^{2}+19u^{2}v^{3}x^{2}z^{2}$$+9u^{2}v^{3}xy^{3}+13uv^{4}xyz^{2}+2uv^{4}y^{4}+4v^{5}y^{2}z^{2})\neq 0$

We also see that $Hess^{(2)}F=0$ , the algebra
property.

$A$.

does not have the weak Lefschetz

24

Example 5.3. The following example is due to Ikeda [5]. Let us choose
the polynomial $F=w^{3}xy+wx^{3}z+y^{3}z^{2}$ . Then the corresponding algebra$A=Q/Ann_{Q}F$ has the Hilbert function (1,4, 10,10,4, 1). The Hessian isgiven as follows:$HessF=8(3w^{7}xy^{4}+8w^{6}x^{6}-27w^{5}x^{3}y^{3}z+27w^{4}y^{6}z^{2}$$-45w^{3}x^{5}y^{2}z^{2}-54w^{2}x^{2}y^{5}z^{3}+9wx^{7}yz^{3}+27x^{4}y^{4}z^{4})$.

In this case, we again have $Hess^{(2)}F$.
Remark 5.1. It is still open whether the Artinian Gorenstein algebra with$\dim A_{1}=3$ has the strong (or weak) Lefschetz property.

References, Cambridge Studies in[1] W. Bruns and J. Herzog, Cohen-Macauley
Advanced Mathematics, 39, Cambridge Univ. Press, Cambridge, 1993.
$7^{\cdot}ings$[2] A. V. Geramita, Inverse systems offat points: Wantng’s problem, secant
varieties of Veronese vanteties and pammeter spaces for Gorenstein ideals, in The Curves Seminar at Queen’s, Vol. X, Queen’s Papers in Pureand Appl. Math., 102, Queen’s Univ., Kingston, ON, 1996, 2-114.
[3] A. V. Geramita, T. Harima, J. C. Migliore and Y. S. Shin, The Hilbert
function of a level algebra, Mem. Amer. Math. Soc. 186 (2007), no. 872,
139 pp.
[4] S. Goto and K. Watanabe, On graded rings I, J. Math. Soc. Japan 30(1978), 179-213.

[5] H. Ikeda, Results on Dilworth and Rees numbers of Artinian local wings,
Japan J. Math. 22 (1996), no. 1, 147-158.
[6] C. R. McDaniel, The strong Lefschetz property for coinvariant rings offinite reflection groups, math. $AC/0909.4184$ .
[7] T. Maeno, Y. Numata and A. Wachi, Strong Lefschetz elements of the
coinvariant rings of finite Coxeter groups, preprint, math.RT/0809.3558.

25[8] T. Maeno and J. Watanabe, Lefschetz elements of Artinian Gorenstein
algebms and Hessians of homogeneous polynomials, math. CA/0903.3581(to appear in Illinois J. Math.)[9] T. Oda, Lectures on torus embeddings and applications (Based on joint
work with Katsuya Miyake), Tata Inst. of Fund. Research 58, SpringerVerlag, Berlin, Heidelberg, New York, 1978.

[10] L. Smith, Polynomial $Inva7\dot{\eta}ants$ of Finite Groups, Research Notes inMathematics Vol. 6, A K Peters Ltd., 1995.

[11] J. Watanabe, A remark on the Hessian of homogeneous polynomials, in
The Curves Seminar at Queen’s Volume XIII, Queen’s Papers in Pureand Appl. Math., Vol. 119, 2000, 171-178.