Special Colloquium

Sándor Kovács (University of Washington) - Vanishing theorems and rational singularities. Rational singularities play an important role in many parts of algebraic geometry. Their most significant property is that their cohomology theory works very much as if they were regular, but the class of rational singularities is much more robust than that of regular points. Traditionally, the two fundamental pillars of studying rational singularities over the complex numbers have been: (i) resolutions of singularities, and (ii) Kodaira-type vanishing theorems. In positive characteristic, however, resolutions of singularities may not exist and Kodaira-type vanishing theorems generally fail to hold. In this talk, I will describe a new approach to rational singularities which do not rely on resolutions as well as a vanishing theorem that is general enough to prove a characteristic independent version of Kempf's criterion for rational singularities. In turn, this result may be used to prove a characteristic independent version of Elkik's theorem which states that most of the singularities of the minimal model program are rational. Another application is to counting rational points on varieties defined over a finite field. In particular, I will discuss a generalization of Esnault's theorem on rational points of smooth Fano varieties to mildly singular log Fano varieties which also gives a new proof of Esnault's theorem.

Talks on Saturday and Sunday

Margarita Castañeda (Posgrado Conjunto UNAM-UMSNH) - Semistable fibrations over $\mathbb{P}^1$ with five singular fibres. We consider a non-isotrivial semistable fibration over the projective line with five singular fibres. In addition, we suppose that the fibration is expressed as the pullback of a pencil on a minimal surface. We will show that $(K_X+F)^2=0$ whenever the genus of the general fibre is sufficiently large (large enough).

John Kopper (UIC) - Effective cycles on blow-ups of Grassmannians. We study the pseudoeffective cones of blow-ups of Grassmannians at sets of points. For small numbers of points, the cones are often spanned by proper transforms of Schubert classes. In some special cases, we provide sharp bounds for when the Schubert classes fail to span and we describe the resulting geometry.

Takumi Murayama (University of Michigan) - Characterizations of projective space and Seshadri constants in arbitrary characteristic. In 1979, Mori showed that projective space is the only smooth variety whose tangent bundle is ample, settling conjectures by Frankel (1961) and Hartshorne (1970). Mori and Mukai conjectured that in fact, a weaker positivity condition should suffice: projective space should be the only n-dimensional Fano variety whose anti-canonical bundle has degree at least n+1 along every curve. While the characteristic zero version of this conjecture was proved by Cho, Miyaoka, and Shepherd-Barron in 2002, it remains open in positive characteristic. We will present some progress in this direction in positive characteristic by giving another characterization of projective space using Seshadri constants and the Frobenius morphism. The key ingredient is a positive-characteristic proof of Demailly's criterion for separation of higher-order jets by adjoint bundles.

Scott Mullane (Harvard University) - The strata of abelian differentials and extremal effective divisors in $M_{g,n}$. An abelian differential defines a flat metric with singularities at its zeros and poles, such that the underlying Riemann surface can be realized as a polygon whose edges are identified pairwise via translation. A number of questions about geometry and dynamics on Riemann surfaces reduce to studying the strata of abelian differentials with prescribed number and multiplicities of zeros and poles.

In this talk we will focus on the divisorial strata closures that form special codimension-one subvarieties in the Deligne-Mumford compactified moduli space of Riemann surfaces. For genus g>1 curves with n>g marked points, we show that infinitely many of these divisors form extremal rays of the cone of effective divisors. Hence these effective cones are not rational polyhedral.

Jun Yong Park (University of Minnesota) - Topology & Arithmetic of moduli space for elliptic Lefschetz fibrations. We consider the moduli space for holomorphic elliptic Lefschetz fibrations over $\mathbb{P}^1$ with nodal singular fibers (fishtails and necklaces) and a distinguished section. Looking at the moduli of weighted projective embeddings, we compute the cardinality of the set of rational points of the moduli space over the finite fields. In the end, we pass the arithmetic invariant of the moduli for elliptic Lefschetz fibrations through the function fields & number fields analogy (global fields analogy) which renders conjectural asymptotic on the ordering of semistable elliptic curves with squarefree conductor.

Neriman Tokcan (University of Illinois at Urbana-Champaign) - A Lower Bound for the Waring Rank. The $K$-rank of a binary form $f$ in $K[x,y],~K\subseteq \mathbb{C},$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We provide lower bounds for the $\mathbb{C}$-rank (Waring rank) and for the $\mathbb{R}$-rank (real Waring rank) of binary forms depending on their factorization. We study binary forms with unique $\mathbb{C}$-minimal representation.

Xiping Zhang (Florida State University) - Characteristic Classes over Determinantal Varieties. Determinantal Varieties are quite interesting varieties, in this talk we give an algorithm to compute the chern-mather class, chern-schwartz-macpherson class, and the local euler obstruction. We also compute the equivariant version of the characteristic classes, with the natural action from the general linear group. From the arithmetic result one can find some interesting results that requires explanations from geometry. See slides here.