Warwick Algebraic Geometry Seminar

Abstracts

Nathan Ilten (Simon Fraser University) - Fujita's Freeness Conjecture for Complexity-One T-Varieties

Given an abstract projective variety X, how do you find a embedding of this variety in projective space? The standard approach is use a divisor to produce a collection of rational functions which induce a rational map to projective space. If the divisor has sufficiently nice properties, this map will be a morphism, or even an embedding. Fujita's freeness conjecture provides a conjectural criterion guaranteeing that this map is indeed a morphism. In this talk, I will provide an overview of Fujita's conjecture, and discuss recent joint work with Klaus Altmann in which we show this conjecture is true whenever X admits a faithful action by a torus of codimension 1.

Miles Reid (Warwick) - Topics in Algebraic Geometry

I will describe the Tate-Oort group Cp of order p in mixed characteristic p, its representations and invariants. These have applications to constructing surfaces and 3-fold with \(Pic^0\) containing p-torsion. The first talk mainly deals in elliptic curves of degree 3 and 5, in preparation for Godeaux surfaces and CY 3-folds with 5-torsion and Campedelli surfaces and CY 3-folds with 7-torsion.

Luca Battistella (Imperial College London) - Genus one reduced invariants of the quintic threefold from maps with cusps

Moduli spaces of stable maps to projective space are not equidimensional in higher genus; there are entire boundary components of degenerate maps. The situation is better understood in genus one: work of J. Li, R. Vakil and A. Zinger led to a desingularization of the main component (the generic element of which represents a map from a smooth elliptic curve) and the introduction of reduced invariants for complete intersections. For threefolds these are related to ordinary Gromov-Witten invariants by the Li-Zinger's formula; they have a better enumerative meaning, discarding a degenerate contribution from rational curves. More recently M. Viscardi has introduced smaller compactifications by allowing maps from Smyth's singularities, e.g. cusps. In joint work with F. Carocci and C. Manolache we show that reduced and cuspidal invariants coincide for the quintic threefold. The rather technical proof uses p-fields and local equations in order to split the intrinsic cone, adapting techniques of H-L. Chang, Y. Hu and J. Li.

Diletta Martinelli (University of Edinburgh) - On the geometry of contractions of the Moduli Space of sheaves of a K3 surface

I will describe how recent advances have made possible to study the birational geometry of hyperkaehler varieties of K3-type using the machinery of wall-crossing and stability conditions on derived categories as developed by Tom Bridgeland. In particular Bayer and Macrì relate birational transformations of the moduli space M of sheaves of a K3 surface X to wall-crossing in the space of Bridgeland stability conditions Stab(X). I will explain how it is possible to refine their analysis to give a precise description of the geometry of the exceptional locus of any birational contractions of M.

Roberto Svaldi (University of Cambridge) - On the boundedness of Calabi-Yau varieties in low dimension

I will discuss new results towards the birational boundedness of low-dimensional elliptic Calabi-Yau varieties, joint work with Gabriele Di Certo. Recent work in the minimal model program suggests that pairs with trivial log canonical class should satisfy some boundedness properties. I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are indeed log birationally bounded. This implies birational boundedness of elliptically fibered Calabi-Yau manifolds with a section, in dimension up to 5. If time allows, I will also try to discuss a first approach towards boundedness of rationally connected CY varieties in low dimension (joint with G. Di Cerbo, W. Chen, J. Han and, C. Jiang).

Francesca Carocci (Imperial College London) - Katz invariants and spectral curves

The genus 0 Gromov- Witten invariants of a smooth projective CY 3-fold X need not to be integers. Conjecturally, the associated integer invariants are DT- type invariants arising from the moduli space of stable 1-dimensional sheaves on X. Using a new technique inspired to Hitchin ‘s spectral curve, we want to give an interpretation of the (local) Katz invariants in terms of maps from curves of genus 0.

Fei Xie (University of Bielefeld) - Non-split toric varieties

I will introduce toric varieties over arbitrary fields and classify minimal smooth projective toric surfaces. I will also give an overview of the Merkurjev-Panin category of "K-motives" and give an explicit construction of a decomposition of certain toric varieties as K-motives into products of central simple algebras. Time permitting, I will discuss the relationship between the decomposition of the K-motive and a semiorthogonal decomposition of the derived category.

Norbert Hoffmann (University of Limerick) - Universal torsors over degenerating del Pezzo surfaces

Let S be a split family of del Pezzo surfaces over a discrete valuation ring such that the general fiber is smooth and the special fiber has ADE-singularities. Let G be the reductive group given by the root system of these singularities. We construct a G-torsor over S whose restriction to the generic fiber is the extension of structure group of the universal torsor under the Neron-Severi torus. This extends a construction of Friedman and Morgan for individual singular del Pezzo surfaces. It is joint work with Ulrich Derenthal.

Junliang Shen (ETH Zurich) - Special subvarieties in holomorphic symplectic varieties

By a result of Bogomolov and Mumford, every projective K3 surface contains a rational curve. Holomorphic symplectic varieties are higher dimensional analogs of K3 surfaces. We will discuss a conjecture of Voisin about algebraically coisotropic subvarieties of holomorphic symplectic varieties, which can be viewed as a higher dimensional generalization of the Bogomolov-Mumford theorem. We show that Voisin's conjecture holds when the holomorphic symplectic variety arises as a moduli space of sheaves on a K3 surface. Finally, we discuss the structure of rational curves in holomophic symplectic varieties. Based on joint work with Georg Oberdieck, Qizheng Yin, and Xiaolei Zhao.

Cristina Manolache (Imperial College London) - A splitting of the virtual class

In order to define intersection theory on spaces with several components of different dimensions one needs to define a "virtual class". I will explain the construction of the virtual class in simple examples and then show how to split this class on the components of the space. I will discuss applications to genus one moduli spaces of stable maps.

Alberto Calabri (University of Ferrara) - On the contractibility of curves on a rational surface

An irreducible plane curve C is said to be contractible is there exists a plane Cremona transformation which maps C to a point. I will first review the classification of irreducible contractible plane curves. I will then deal with the case of reduced, reducible contractible plane curves, in particular of the unions of many lines in the plane. More generally, I will state the few known results concerning the contractibility of reduced curves on a rational surface and I will report on a joint work in progress with Ciro Ciliberto regarding contractibility theorems.