# Warwick Algebraic Geometry Seminar

### Summer Term 2018

The Warwick Algebraic Geometry Seminar will be taking place this term on Tuesday afternoons at 2pm in MS.05, unless otherwise stated. We also have a later slot available to us on Tuesdays at 4pm in B3.03, which we may make use of occasionally.

In addition to our own activities, we will also be arranging regular trips to various algebraic geometry activities running in the UK, including the COW seminar, the East Midlands Seminar in Geometry (EmSG), the London Geometry and Topology Seminar, the GLEN seminar, and the British Algebraic Geometry meeting (BrAG).

If you are interested in receiving announcements about upcoming seminars and other algebraic geometry activities at Warwick, you're welcome to join our mailing list. To do this, just send an email to Liana Heuberger (l.heuberger (at) warwick.ac.uk) or Christian Böhning (c.boehning (at) warwick.ac.uk) and ask to be added to the list.

 Week Date Speaker Title 1 24th April Nathan Ilten (MS.05) Fujita's Freeness Conjecture for Complexity-One T-Varieties 1 24th April Miles Reid (B3.03) Topics in Algebraic Geometry 2 1st May Miles Reid (B3.03) Topics in Algebraic Geometry 3 8th May Luca Battistella (MS.05) Genus one reduced invariants of the quintic threefold from maps with cusps 3 8th May Miles Reid (B3.03) Topics in Algebraic Geometry 4 15th May Nikolai Tyurin (MS.05) Special Bohr-Sommerfeld Geometry of Algebraic Varieties 5 22nd May Diletta Martinelli (MS.05) On the geometry of contractions of the Moduli Space of sheaves of a K3 surface 5 22nd May Roberto Svaldi (B3.03) On the boundedness of Calabi-Yau varieties in low dimension 6 29th May Francesca Carocci (MS.05) Katz invariants and spectral curves 7 5th June Fei Xie (MS.05) Non-split toric varieties 8 12th June Norbert Hoffmann (MS.05) Universal torsors over degenerating del Pezzo surfaces 9 19th June Junliang Shen (MS.05) Special subvarieties in holomorphic symplectic varieties 9 19th June Cristina Manolache (B3.03) A splitting of the virtual class 10 26th June Alberto Calabri (MS.05) On the contractibility of curves on a rational surface

Details of last term's seminars may be found here.

## 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.

## Getting Here

Directions to the university may be found here. Once you're on campus, the Mathematics Institute is located in the Zeeman building; you can download a map of the campus here.

Please note that if you are arriving by public transport, the University of Warwick is not in fact in the town of Warwick, or indeed anywhere near it. Instead, it is located a short distance southwest of Coventry. If you are coming by train the closest stations are Coventry and Leamington Spa.

To get to campus from Coventry station you should take bus 11, 11U, or 12X; all three leave from stand ER3 at the bus hub outside the railway station. At the time of writing, a single ticket from Coventry station to the university costs £2.10; a day ticket is £4; please note that the buses from Coventry only accept exact change.

To get to campus from Leamington Spa station you should take bus U1, U2, or U17. Please note that these buses do not leave from directly outside the station; instead, the nearest bus stop is just around the corner on Victoria Terrace. A map of the route may be found here. At the time of writing, a single ticket from Leamington Spa station to the university costs £2.75.