# Warwick Algebraic Geometry Seminar

### Autumn Term 2017

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) TBA 7 5th June Fei Xie (MS.05) TBA 8 12th June Norbert Hoffmann (MS.05) TBA 9 19th June Junliang Shen (MS.05) TBA 10 26th June Alberto Calabri (MS.05) TBA

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 Marinelli (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).

## 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.