Mathematisches Forschungsinstitut Oberwolfach

Report No. 2

Moduli Spaces in Algebraic Geometry
Organised by
Dan Abramovich, Providence
Gavril Farkas, Berlin
Stefan Kebekus, Freiburg

January 10th – January 16th, 2010
Abstract. Invariants of topological spaces of dimension three play a majorrole in many areas, in particular . . .
Mathematics Subject Classiﬁcation (2000): AMS-CLASSIFICATION.

Introduction by the Organisers
The workshop Invariants of topological spaces of dimension three, organised by
Max Muster (München) and Bill E. Xample (New York) was held March 1st–March 6th, 2005. This meeting was well attended with over 30 participants withbroad geographic representation from all continents. This workshop was a niceblend of researchers with various backgrounds . . .

Moduli Spaces in Algebraic Geometry

3Workshop: Moduli Spaces in Algebraic Geometry
Table of Contents
Shigeru Mukai
Polarized K3 surfaces of gnenus 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5Fréderic Franco̧is Déchamps, Peter Mustermann
Computing other invariants of topological spaces of dimension three . . . . .

5Moduli Spaces in Algebraic Geometry

5Abstracts
Polarized K3 surfaces of gnenus 16Shigeru Mukai
Let T = G(2, 3; C4 ) be the EPS-moduli space of the twisted cubics in P3 constructed in [1]. T is the GIT-quotient of C2 ⊗C3 ⊗C4 by the action of GL(2)×GL(3)on the ﬁrst and second factors. There exist two tautological vector bundles E, Fof rank 3, 2 and the universal homomorphism E ⊗ C4 −→ F on T . The vector
bundle E embeds T into the 21-dimensional Grassmannian G(S 2 C4 , 3).
Theorem (1) A general complete intersection S with respect to the rank 10 vector
bundle E ⊕2 ⊕ F ⊕2 in the EPS-moduli space T is a K3 surface, and det E|S is apolarization of genus 16, that is, degree 30.
(2) Moreover, a moduli-theoretically general polarized K3 surface (S, h) of genus
16 is obtained in this way.
Let Fg be the moduli space of primitively (quasi-)polarized K3 surfaces (S, h)of degree 2g − 2, and Sg be the (quasi-)universal family over it. The theoremyields a dominant rational map P 36 · · · → F16 from a G(2, 12)-bundle P 36 overthe 16-dimensional Grassmannian G(2, S 2 C4 ) of pencils of quadrics to F16 .
Corollary The moduli space F16 is unirational.
See [2] and [3] for the birational type of other Fg ’s.
Since E|S is a stable semi-rigid vector bundle with Mukai vector v = (3, h, 5),
the rational map factors through S16 .
Conjecture The induced rational map P 36 //P GL(4) · · · −→ S16 between 21dimensional varieties is birational.
References[1] G. Ellingsrud, R. Piene, and S.A. Strømme, On the variety of nets of quadrics deﬁning
twisted cubic curves. In F. Ghione, C. Peskine and E. Sernesi, editors, Space Curves, Lecture
Notes in Math. 1266 (1987), Springer-Verlag, pp. 84–96.
[2] V.A. Gritsenko, K. Hulek and G.A. Sankaran, The Kodaira dimension of the moduli spacesof K3 surfaces, Invent. math. 169 (2007), 519–567.
[3] S. Mukai, Polarized K3 surfaces of genus thirteen, In S. Mukai et. al, editors, Moduli spaces
and Arithmetic Geometry (Kyoto, 2004), Adv. Stud. Pure Math. 45 (2006), Math. Soc.
Japan and Amer. Math. Soc., pp. 315–326.

Computing other invariantsof topological spaces of dimension threeFréderic Franco̧is Déchamps, Peter Mustermann
The computation of ...

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Oberwolfach Report 2
Reporter: Daniel Greb