# Summer Term 2017

Shunsuke Takagi (University of Tokyo). General hyperplane sections of 3-folds in positive characteristic. Friday 5th May, 1:30-2:30pm. Huxley 145.

Abstract: Miles Reid proved that in characteristic zero, a general hyperplane section of a canonical (resp. klt) 3-hold has only rational double points (resp. klt) singularities. Since his proof heavily depends on the Bertini theorem for free linear series, it is not clear whether a similar statement holds in positive characteristic or not. In this talk, I will present an affirmative answer to this question when the characteristic is larger than 5. This is joint work with Kenta Sato.

Ivan Smith (University of Cambridge). Lagrangian pinwheels and Markov numbers. Friday 12th May, 1:30-2:30pm. Huxley 145.

Abstract: Pinwheels are two-dimensional cell complexes obtained by attaching a disc to a circle by a map of degree at least 2. We will discuss Lagrangian embedding questions for pinwheels, concentrating on the case of pinwheels in the complex projective plane. This talk reports on joint work with Jonny Evans.

Karl Schwede (University of Utah). Using perfectoid algebras to study singularities in mixed
characteristic and applications.
Friday 19th May, 1:30-2:30pm. Huxley 145.

Abstract: The multiplier ideal has been a key tool for studying singularities of higher dimensional complex algebraic varieties over the past decades. In characteristic p > 0, there is an analog of the multiplier ideal called the test ideal which first appeared in commutative algebra. In this talk I first will review the characteristic zero and positive characteristic theories. Then, building upon recent ideas using perfectoid algebras and spaces in the proof of the (derived) direct summand conjecture by Andre and Bhatt, I will discuss a mixed characteristic theory which captures similar information. As an application, we obtain a bound on the growth of symbolic powers of prime ideals in regular rings (analogous to an application of multiplier/test ideals in characteristic 0/p). This is joint work with Linquan Ma.

Andreas Gross (Imperial College). Chi-y genera of generic intersections in algebraic tori and refined tropicalizations. Friday 26th May, 1:30-2:30pm. Huxley 145.

Abstract: An algorithm to compute chi-y genera of generic complete intersections in algebraic tori has already been known since the work of Danilov and Khovanskii in 1978, yet a closed formula has been given only very recently by Di Rocco, Haase, and Nill. In my talk, I will show how this formula simplifies considerably after an extension of scalars. I will give an algebraic explanation for this phenomenon using the Grothendieck rings of vector bundles on toric varieties. We will then see how the tropical Chern character gives rise to a refined tropicalization, which retains the good properties of the usual, unrefined tropicalization.

De-Qi Zhang (National University of Singapore). Polarized endomorphisms of projective varieties. Friday 2nd June, 1:30-2:30pm. Huxley 145.

Abstract: An endomorphism f on a normal projective variety X is polarized if the f-pullback of an ample divisor H on X is linearly equivalent to the multiple qH for some natural number q larger than 1. Examples of such f include self-maps of the projective spaces (or more generally Fano varieties of Picard number 1) and multiplication map of complex tori. We show that we can run the f-equivariant minimal model program (MMP) on smooth or mildly singular X, and conclude that the building blocks of polarized endomorphisms are those on Fano varieties or complex tori and their quotients. This is a joint work with S. Meng.

No seminar on Friday 9th June due to this event.

Pierrick Bousseau (Imperial College). Tropical refined curve counting from Gromov-Witten invariants. Friday 16th June, 1:30-2:30pm. Huxley 145.

Abstract: I will start reviewing Mikhalkin’s correspondence theorem between counts of complex algebraic curves in toric surfaces and counts of tropical curves in the real plane. I will then present a new geometric interpretation of the refined counts of tropical curves introduced by Block and Göttsche. This will involve generating series of log Gromov-Witten invariants and a mysterious change of variables.

Cameron Gordon (University of Texas at Austin). Left-orderability, L-spaces, and cyclic branched coverings. Friday 23rd June, 1:30-2:30pm. Huxley 145.

Abstract: It has been conjectured that for a prime 3-manifold $M$ the following are equivalent: (1) $\pi_1(M)$ is left-orderable, (2) $M$ admits a co-orientable taut foliation, and (3) $M$ is not a Heegaard Floer L-space. We will discuss this in the special case where $M$ is the $n$-fold cyclic branched cover of a knot.