# 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 3rd October Christian Böhning (MS.05) The $$p$$-torsion in the Brauer group in characteristic $$p$$ as an obstruction to Chow zero universal triviality and rationality questions 2 10th October Daniel Greb (MS.05) Differential forms on singular varieties with trivial canonical class 3 17th October Sara Muhvić (MS.O5) G-Hilb and crepant resolutions of certain abelian orbifolds in dimension 4 3 17th October Tom Ducat (MS.B3.03) The decomposition groups of plane conic and rational cubic curves 5 31st October Pierrick Bousseau (MS.05) Quantum mirrors of log Calabi-Yau surfaces and higher genus curves counting 6 7th November Matthias Schütt (MS.05) Zariski K3 surfaces 7 14th November New arrivals week (MS.05) - 8 21st November Evgeny Shinder (MS.05) Specialization of (stable) rationality 8 21st November Gergö Pintér (B3.03) The boundary of the Milnor fibre of certain non-isolated singularities 9 28th November Artie Prendergast-Smith (MS.05) Effective cycles in higher codimension 9 28th November Isac Hedén (B3.03) Extensions of principal additive bundles over a punctured surface 10 5th December Alessandro Bigazzi (MS.05) Degeneration of conic bundles in characteristic 2 and stable rationality 10 5th December Miles Reid (B3.03) Topics in Algebraic Geometry 11 12th December DongSeon Hwang (MS.05) Log del Pezzo Surfaces of Picard Number One

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

## Abstracts

Christian Böhning (University of Warwick) - The $$p$$-torsion in the Brauer group in characteristic $$p$$ as an obstruction to Chow zero universal triviality and rationality questions
We establish the $$p$$-torsion in the Brauer group of a smooth projective variety over a field of characteristic $$p$$ as an obstruction to universal $$CH_0$$-triviality of that variety. For $$p=2$$, we compute this for some (desingularisations of) conic bundles over $$ℙ^2$$ in terms of their discriminant profiles. Using the degeneration method by Voisin-Colliot-Thelene-Pirutka-Totaro et al. in an unequal characteristic set-up, this yields new explicit examples of threefold conic bundles, defined over $$ℤ$$, that are not stably rational (over $$ℂ$$). We believe this can be extended to yield information for fourfold conic bundles as well, and in particular for Gushel-Mukai fourfolds. This is joint work with Asher Auel, Alessandro Bigazzi and Hans-Christian von Bothmer.
Daniel Greb (Essen University) - Differential forms on singular varieties with trivial canonical class
The famous "decomposition theorem" asserts that any projective manifold with vanishing first Chern class admits an étale cover that decomposes as a product of an Abelian variety, simply-connected Calabi-Yau manifolds, and irreducible holomorphic-symplectic (IHS) manifolds (also called hyperkähler manifolds). The three classes can be distinguished by looking at their algebra of holomorphic differential forms. In the Minimal Model Program, we are forced to work with singular varieties. I survey recent progress towards a decomposition in the relevant singular case, and argue that an analogue of the trichotomy "Abelian / Calabi-Yau / IHS" exists in this more general setting. The talk is based on joint work with Henri Guenancia and Stefan Kebekus.
Sara Muhvić (University of Warwick) - G-Hilb and crepant resolutions of certain abelian orbifolds in dimension 4
Let G ⊂ SL(4, $$ℂ$$) be a maximal diagonal subgroup of exponent $$r$$. In this talk, I will describe the scheme G-Hilb($$ℂ^4)$$ through its toric fan and show how it can be used to construct "special" crepant resolutions of the quotient variety $$ℂ^4∕G$$.
Tom Ducat (University of Bristol) - The decomposition groups of plane conic and rational cubic curves
The decomposition group of a plane curve C is the subgroup of elements of the Cremona group which restrict to a birational map of C. Hedén and Zimmermann proved that when C is a line then this group is generated by linear maps and one elementary quadratic map preserving C - an analogy of the Noether-Castelnuovo theorem for the full Cremona group. Using this result we show that the decomposition group is also generated by linear and quadratic maps when C is a conic or rational cubic curve, but not in general for plane rational curves of degree ≥4. This is joint work with Isac Hedén and Susanna Zimmermann.
Pierrick Bousseau (Imperial College London) - Quantum mirrors of log Calabi-Yau surfaces and higher genus curves counting
I will start explaining a correspondence theorem between Block-Göttsche refined tropical curves counting and some higher genus log Gromov-Witten invariants of toric surfaces (reference: arxiv 1706.07762). I will then describe an application of this result to a synthetic construction of noncommutative deformations of Gross-Hacking-Keel mirror families of log Calabi-Yau surfaces.
Matthias Schütt (Leibniz University at Hannover) - Zariski K3 surfaces
It is a classical fact in algebraic geometry that unirational curves are rational, and the same holds true for surfaces in characteristic zero, but not in positive characteristic or in higher dimension. In this talk I will report on joint work with T. Katsura to construct Zariski K3 surfaces, i.e. K3 surfaces admitting a purely inseparable map of degree p from the projective plane. In particular, we will prove that any supersingular Kummer surface is Zariski in certain characteristics.
Evgeny Shinder (University of Sheffield) - Specialization of (stable) rationality
The specialization question for rationality is the following one: assume that very general fibers of a flat proper morphism are rational, does it imply that all fibers are rational? I will talk about recent solution of this question in characteristic zero due to myself and Nicaise, and Kontsevich-Tschinkel. The method relies on a construction of various specialization morphisms for the Grothendieck ring of varieties (stable rationality) and the Burnside ring of varieties (rationality).
Gergö Pintér (Rényi Alfréd Mathematics Institute, Budapest) - The boundary of the Milnor fibre of certain non-isolated singularities
We determine the boundary of the Milnor fibre of those non-isolated hypersurface surface singularities, which can be parametrized with finitely determined map-germs.
Artie Prendergast-Smith (Loughborough University) - Effective cycles in higher codimension
We will survey some recent results of others about cycles of codimension higher than 1, including differences with the case of divisors and examples where cones of effective cycles can be computed. We will then explain joint work with Norbert Pintye about blowups of lines in projective space.
Isac Hedén (University of Warwick) - Extensions of principal additive bundles over a punctured surface
We study complex affine $$Ga$$-threefolds $$X$$ such that the restriction of the quotient morphism $$\pi\colon X\to S$$ to $$\pi^{-1}(S_*)$$ is a principal $$Ga$$-bundle, where $$S_*$$ denotes the complement of a closed point in $$S$$ and $$Ga$$ denotes the additive group over the field of complex numbers. Changing the point of view, we look for affine extensions of $$Ga$$-principal bundles over punctured surfaces, i.e affine varieties that are obtained by adding an extra fiber to the bundle projection over the puncture. The variety $$SL_2$$ will be of special interest and a source of many examples.
Alessandro Bigazzi (University of Warwick) - Degeneration of conic bundles in characteristic 2 and stable rationality
We exhibit an explicit application of the degeneration method by Voisin, Colliot-Thélène and Pirutka in the mixed characteristic case, showing the non-triviality of the 2-torsion of the unramified Brauer group for a specific conic bundles defined over an algebraically closed field of characteristic 2 by means of its discriminant profile. Finally, we survey some of the possible further developments of this method. This is joint work with A. Auel, Ch. Boehning and H.C. Graf von Bothmer.
DongSeon Hwang (Ajou and Warwick) - Log del Pezzo surfaces of Picard number one
There have been numerous attempts to classify log del Pezzo surfaces. In this talk, I will quickly summarize the known attempts at this goal and report my recent work on the classification of log del Pezzo surfaces of Picard number one. The result is obtained by generalizing the notion of 'cascades' of nonsingular del Pezzo surfaces, based on the approach initiated by Miyanishi and Zhang. Several explicit examples will be demonstrated in DP1 calculator, a website under construction. If time permit, I will present an example of Kollar which suggests some relationship between my cascades and Reid’s cascades.

## 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.