Mathematisches Forschungsinstitut Oberwolfach

Report No. 7/2005

Workshop: Komplexe Algebraische Geometrie
Organised byFabrizio Catanese (Bayreuth)
Yujiro Kawamata (Tokyo)
Gang Tian (MIT)Eckart Viehweg (Essen)

February 13th – February 19th, 2005

Introduction by the Organisers
Something meaninful...
MSC Classiﬁcation:

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Oberwolfach Report 7/2005

Workshop: Komplexe Algebraische Geometrie

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Workshop: Komplexe Algebraische Geometrie
Table of Contents
Shigeru Mukai
Geometric proof of ﬁnite generation of certain rings of invariants . . . . . . . . 5

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Workshop: Komplexe Algebraische Geometrie

5Abstracts
Geometric proof of ﬁnite generation of certain rings of invariants
Shigeru MukaiLet ρ : Cn ↓ S = C[x1 , . . . , xn , y1 , . . . , yn ] be the standard unipotent linear
action of the n-dimensional additive group Cn on the polynomial ring S of 2nxi → xi
variables, that is, (t1 , . . . , tn ) ∈ Cn acts byfor 1 ≤ i ≤ n. Inyi → yi + ti xi
1958, Nagata[5] proved that the ring of invariants S G with respect to a generallinear subspace G ⊂ Cn of codimension 3 was not ﬁnitely generated for n = 16.
We studied this example more systematically and obtained the following:
Theorem The ring of invariants S G of ρ with respect to a general linear subspace
G ⊂ Cn of codimension r is ﬁnitely generated if and only if
1 1
1+ +> 1.
2 r
n−rThis inequality is equivalent to the ﬁniteness of the Weyl group of the Dynkin
diagram T2,r,n−r with three legs of length 2, r and n − r. The ‘only if’ part of thetheorem follows from this observation and the following geometric interpretationof S G . (See [2] for the details.)
Proposition Let Pr−1 be the projective space P∗ (Cn /G) and {p1 , . . . , pn } ⊂Pr−1 be the image of the standard basis of Cn . Then the ring of invariants S G
is isomorphic to the total coordinate ring, or the Cox ring, T C(X) of the blow-up
X = XG of Pr−1 at the n points p1 , . . . , pn .
In this talk, as a continuation of [4], I explained the proof of ‘if’ part in the casedim G = 2. The variety X in the proposition is the blow-up of Pn−3 at n points ingeneral position. The key of our proof is that X is the moduli space of parabolic
2-bundles over an n-pointed projective line (P1 : p1 , . . . , pn ) for a certain weight.
This fact enables us to determine the eﬀective cone of X, the movable cone Mov X(see [2]) and its chamber structure. For example, taking
i−KX = (n − 2)h − (n − 4) 1 ei f1 = h + e1 − e2 − · · · − en
f2 = h − e1 + e2 − · · · − en.
 ..
fn = h − e1 − e2 − · · · + en
nas a basis of Pic X ⊗ Q, a divisor D ∼ −aKX + 1 bi fi is movable if and only if(n − 4)a −bi +bj ≥ 0i∈Ij∈Iholds for every I ⊂ {1. . . . , n} with |I| even and |bi | ≤ a holds for evey 1 ≤ i ≤ n,
where h is the pull-back of a hyperplane and e1 , . . . , en are the exceptional divisors.

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The cone Mov X is divided into ﬁnitely many chambers, which are rational
polyhedral cones, by the ﬂopping walls. For every movable divisor D on X, thegraded ring n≥0 H 0 (X, nD) is ﬁnitely generated by the GIT-construction of themoduli spaces. Hence the (Mov X)-part of T C(X) is ﬁnitely generated. The totalcoordinate ring T C(X) is ﬁnitely generated since it is generated by the equationsof 2n−1 exceptional divisors, whose linear equivalence classes are1eI = (−KX −fi +fj )4i∈Ij∈Iwith I odd, over the (Mov X)-part. In the case of dim G = 3 (and n ≤ 8), theﬁnite generation is similarly proved replacing the parabolic bundles over P1 byvector bundles over a del Pezzo surface.
In the case r = 3, the minimal (ﬁnite) set of generators of T C(X), X itself
being a del Pezzo surface, is determined in [1]. Other cases of ‘if’ part is easy.
References[1] V. Batyrev and O.N. Popov, The Cox ring of a del Pezzo surface, Arithmetic of higherdimensional algebraic varieties, eds. Poonen and Tschinkel, Birkhauser, 2004, pp. 85–103.
[2] Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degeneration of surfaces, Ann. of Math. 127(1988), 93–163.
[3] S. Mukai, Counterexample to Hilbert’s fourteenth problem for three dimensional additive
groups, RIMS preprint, #1343, 2001.
[4] S. Mukai, Finite and inﬁnite generation of Nagata invariant ring, Oberwolfach Reports
2004, Springer Verlag.
[5] M. Nagata, On the fourteenth problem of Hilbert, Proc. Int’l Cong. Math., Edingburgh,1958, pp. 459–462, Cambridge Univ. Press, 1960.

Reporter: Fabio Tonoli