# 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 9th January Miles Reid (B3.03) Topics in algebraic geometry #1 (MA610) 2 16th January Miles Reid (B3.03) Topics in algebraic geometry #2 (MA610) 3 23rd January Alan Thompson (MS.05) ADE Surfaces 3 23rd January Miles Reid (B3.03) Topics in algebraic geometry #3 (MA610) 4 30th January Dhruv Ranganathan (MS.05) The space of equations for a curve of prescribed gonality 5 6th February Navid Nabijou (MS.05) Relative Quasimaps and a Lefschetz-type Formula 5 6th February Miles Reid (B3.03) Horikawa quintics 6 13th February Conference on K-theory week 7 20th February Takeru Fukuoka (MS.05) Relative linear extensions of sextic del Pezzo fibrations 8 27th February 9 6th March Enrica Mazzon (MS.05) Dual complexes of degenerations and Berkovich geometry 10 13th March Summer term 1 24th April Roberto Svaldi (MS.05) TBA 1 24th April Nathan Ilten (B3.03) TBA 2 1st May 3 8th May 4 15th May Nikolai Tyurin (MS.05) TBA 5 22nd May Diletta Martinelli (MS.05) TBA 6 29th May 7 5th June 8 12th June 9 19th June 10 26th June

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

## Abstracts

Miles Reid (Warwick) - Topics in Algebraic Geometry
I will talk about some work I have been involved in over the last few decades. I will not necessarily be concerned with 100% technically correct theoretical statements. I will improvise, but the first couple of sessions will probably be about curves, $$K3$$ surfaces and $$\mathbb{Q}$$-Fano 3-folds.
Alan Thompson (Cambridge) - ADE Surfaces
I will discuss the problem of classifying rational surfaces equipped with both an anticanonical divisor and an involution; such surfaces appear naturally as components of degenerations of K3 surfaces with antisymplectic involution. This problem leads naturally to a class of surfaces that we call ADE surfaces. ADE surfaces have a number of interesting properties, chief among which is a surprising link to root lattices of types ADE (hence the name); such lattices classify the possible families and determine both their moduli and degenerations. This is joint work with Valery Alexeev.
Dhruv Ranganathan (MIT) - The space of equations for a curve of prescribed gonality
The Brill-Noether varieties of a curve C parameterize embeddings of C of prescribed degree into a projective space of prescribed dimension, i.e. equations for the curve. When C is general, these varieties are well understood: they are smooth, irreducible, and have the "expected" dimension. As one ventures deeper into the moduli space, past the general curve, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. Based on an analogous combinatorial problem on graphs, Pflueger conjectured a formula for the dimensions of the Brill-Noether varieties for general curves of a given gonality. I will present joint work with Dave Jensen, in which we prove Pflueger's conjecture. The proof blends ideas fromGromov-Witten theory, logarithmic and tropical geometry, and the theory of Berkovich spaces.
The theory of stable quasimaps is an important tool in enumerative geometry, providing an alternative system of curve counts to the usual Gromov-Witten invariants. In joint work with Luca Battistella, we define moduli spaces of relative stable quasimaps to a pair $$(X,Y)$$, where $$Y$$ is a hyperplane section in X. Intuitively, these spaces parametrise quasimaps to $$X$$ with specified orders of tangency to $$Y$$. We apply this construction to obtain a Lefschetz-type formula, expressing certain quasimap invariants of $$Y$$ in terms of invariants of $$X$$. I will begin this talk with a broad introduction to modern enumerative geometry (including Gromov-Witten theory); I will then explain the philosophy behind relative invariants, before finally discussing our application of this philosophy in the setting of quasimaps.
Miles Reid (Warwick) - Horikawa quintics
A Horikawa quintic is a polarised n-fold $$(X, A)$$ with $$A^n = 5$$, $$h^0(A) = n+2$$ and $$K_X = (3-n)A$$. Assume that $$X$$ is nonsingular, or has mild singularities (for example, at worst ordinary double locus in codimension $$\geq 3$$). One expects initially the linear system $$|A|$$ to be free, defining an isomorphism with a quintic hypersurface $$X = X_5$$ in $$\mathbb{P}^{n+1}$$ (Type I), but there is another possibility: $$|A|$$ may have a single transverse base point $$P$$ in $$X$$, and define a double cover $$X \rightarrow Q$$ to a quadric $$Q$$ in $$\mathbb{P}^{n+1}$$. It turns out that Q can only have rank $$4$$ (Type II$$_a$$) or rank $$3$$ (Type II$$_b$$). The most interesting case is the deformation theory of $$X$$ of Type II$$_b$$: this has small deformations to Type I and to Type II$$_a$$, and for $$n \geq 3$$ these are topologically different (they have different Betti numbers). When $$n = 3$$ this provides an interesting case of "conifold transition".
Takeru Fukuoka (Tokyo) - Relative linear extensions of sextic del Pezzo fibrations
An extremal contraction from a non-singular projective 3-folds onto a smooth curve is so-called a del Pezzo fibration. The degree of a del Pezzo fibration is defined to be the anti-canonical degree of general fibers. It is classically known that every del Pezzo surface $$S$$ is a (weighted) complete intersection of a certain Fano variety. For example, if $$S$$ is of degree 6, then $$S$$ is a hyperplane section of $$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$$ and also a linear section of $$\mathbb{P}^2 \times \mathbb{P}^2$$.In order to study del Pezzo fibrations, it is important to relativize such descriptions for those. In this talk, we will mainly discuss on del Pezzo fibrations of degree 6. One of the main result shows that those are relative linear sections of $$\mathbb{P}^1 \times \mathbb{P}^1 \times\mathbb{P}^1$$-fibrations and $$\mathbb{P}^2 \times \mathbb{P}^2$$-fibrations, which are constructed as Mori fiber spaces with smooth total space. As the application, we will completely classify the singular fibers of sextic del Pezzo fibrations.
Enrica Mazzon (Imperial College London) - Dual complexes of degenerations and Berkovich geometry
To a degeneration of varieties, we can associate the dual intersection complex, a topological space that encodes the combinatoric of the central fiber and reflects the geometry of the generic fiber. The points of the dual complex can be identified to valuations on the function field of the variety, hence the dual complex can be embedded in the Berkovich space of the variety. In this talk I will explain how this interpretation gives an insight in the study of the dual complexes. I will focus on some degenerations of hyper-Kähler varieties and show that we are able to determine the homeomorphism type of their dual complex using techniques of Berkovich geometry. The results are in accordance with the predictions of mirror symmetry, and the recent work about the rational homology of dual complexes of degenerations of hyper-Kähler varieties, due to Kollár, Laza, Saccà and Voisin. This is joint work with Morgan Brown.
Francesca Carocci (Imperial College London) - Reduced vs Cuspidal GW invariants for the quintic 3-fold
Moduli spaces of stable maps of genus g>0 are highly singular and with many irreducible components which affect the enumerative meaning of the invariants arising from them. In this talk we will try to give a flavour of how bad these spaces can be, already in the simplest example in genus 1. We will then hint at two possible approaches to deal with the so called "degenerate contributions", namely: Li-Zinger reduced invariants and Viscardi-Smyth cuspidal invariants. We will then explain in which sense these two approaches coincide for the quintic 3-fold.

## 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.