Algebraic Geometry Seminar

I will give an introduction to Oliver Lorscheid’s theory of ordered blueprints – one of the more successful approaches to “the field of one element” – and sketch its relationship to Berkovich spaces, tropical geometry, Tits models for algebraic groups, and moduli spaces of matroids. The basic idea for the latter two applications is quite simple: given a scheme over Z defined by equations with coefficients in {0,1,-1}, there is a corresponding “blue model” whose K-points (where K is the Krasner hyperfield) sometimes correspond to interesting combinatorial structures. For example, taking K-points of a suitable blue model for a split reductive group scheme G over Z gives the Weyl group of G, and taking K-points of a suitable blue model for the Grassmannian G(r,n) gives the set of matroids of rank r on {1,...,n}. Similarly, the Berkovich analytification of a scheme X over a valued field k coincides, as a topological space, with the set of T-points of X, considered as an ordered blue scheme over k. Here T is the tropical hyperfield, and T-points are defined using the observation that a (height 1) valuation on k is nothing other than a homomorphism to T.

In this talk, we discuss Fano varieties, one of the three building blocks of algebraic varieties. Among them, the most well-known class are Fano toric varieties that can be described combinatorially. In this talk, we will introduce two invariants for Fano varieties: the complexity and coregularity. Both invariants take value 0 when the Fano variety is toric. We will explain how these two invariants allow us to detect toric varieties among Fano varieties, and what is expected for Fano varieties when these invariants are near 0.

Let’s count: 1, 2, q+1. The eponymous objects are special projective hypersurfaces of degree q+1, where q is a power of the positive ground field characteristic. This talk will sketch an analogy between the geometry of q-bic hypersurfaces and that of quadric and cubic hypersurfaces. For instance, the moduli spaces of linear spaces in q-bics are smooth and themselves have rich geometry. In the case of q-bic threefolds, I will describe an analogue of result of Clemens and Griffiths, which relates the intermediate Jacobian of the q-bic with the Albanese of its surface of lines.

Using MacPherson's Euler obstruction function, one can identify the abelian group of constructible functions with the group of algebraic cycles on a smooth complex algebraic variety. Kashiwara's local index formula gives an alternative approach to this identification by using characteristic cycles for holonomic D-modules (they are Lagrangian cycles in the cotangent bundle). This identification then enables us to define Chern classes of algebraic cycles by using characteristic cycles. In this talk, I will first explain how to obtain Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement by using logarithmic cotangent bundles motivated by D-module theory. Then I will discuss applylications of such Chern classes in understanding Chern-Mather classes of very affine varieties and in proving the Involution Conjecture of Huh and Sturmfels in likelihood geometry. This work is joint with Maxim, Rodriguez, and Wang.

Within this talk, which is joint work with Matthew Ballard, we will study the bounded derived category of coherent sheaves $D^b (\operatorname{coh}X)$ on an $F$-finite Noetherian scheme of prime characteristic $p$. It will be shown that $F_\ast^e (\operatorname{perf}X)$ strongly generates $D^b (\operatorname{coh}X)$ for $e\gg 0$ where $F_\ast^e$ denotes the $e$-th iterate of the Frobenius pushforward functor on $X$. Immediate applications to this is a new proof of Kunz's theorem, introduces new invariants for singularity types in prime characteristic coming from derived categories, categorical resolution of singularities, and produces interesting structural insight into this category. This motivates a notion of $F$-thickness; namely, when $F_\ast^e \mathcal{O}_X$ strongly generates $D^b (\operatorname{coh}X)$. Lastly, in the case of smooth Fano projective complete intersections $X$, it will be shown that $F_\ast^e \mathcal{O}_X$ decomposes into direct summands indexed by components in Kuznetsov/Orlov style semiorthogonal decompositions. Particularly, when $p$ is sufficiently large, $X$ is globally $F$-split.

A stable n-marked curve is a nodal curve with n distinct marked points and finitely many automorphisms. If we choose rational numbers a_1, . . ., a_n in the interval (0, 1], then a weighted stable n-marked curve is a generalization where the marks are allowed to coincide as long as the total weight at any point is at most one. Moduli of weighted stable curves were first constructed by Hassett. On the other hand, a twisted stable n-marked curve is a tame stack whose coarse moduli space is a stable n-marked curve, such that stacky structure is concentrated at nodes and markings and has a specific local description. I will discuss a modification (using log geometry) of the moduli of twisted stable curves where the markings are allowed to coincide, analogous to Hassett's construction for representable curves. This is a joint project with Martin Olsson.

Log Fano cones are generalizations of cones over log Fano pairs, and have a local K-stability theory extending the one for log Fano pairs. In this talk, we aim to give a characterization for local K-stability from a non-Archimedean point of view, which will allow us to study general test configurations without putting strong singularity assumptions on them. As a consequence of this characterization, we will see that one direction of the YTD conjecture for log Fano cones, as shown earlier by Collins-Székelyhidi and Li, can be strengthened.

We show that a very general n-dimensional complex hypersurface X of degree ≥ 5⌈(n+3)/6⌉ has no finite order birational automorphisms. In particular, this applies to higher index Fano hypersurfaces (of sufficiently high dimension). We prove this result by studying the specialization homomorphism for the birational automorphism group. This work is joint with Nathan Chen and David Stapleton.

In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Motivated by the study of indecomposable vector bundles, in 1978 Beilinson constructed a resolution of the diagonal on P^n which has been used to great effect in algebraic geometry. We obtain a Horrocks-type splitting criterion (under an additional hypothesis) for vector bundles over a smooth projective toric variety X of Picard rank 2 using a linear resolution of the diagonal consisting of finite direct sums of line bundles. Since this resolution has length dim(X), we also prove a new case of a conjecture of Berkesch--Erman--Smith that predicts a version of Hilbert's Syzygy Theorem for virtual resolutions. This is joint work with Michael Brown.

We show that every Deligne-Mumford gerbe over a field occurs as the residual gerbe of a point of the moduli stack of curves. Informally, this means that the moduli space of curves fails to be a fine moduli space in every possible way. We also show the same result for a list of other natural moduli problems. This is joint work with Max Lieblich.

Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. Starting with the pioneering work of Mark Green on curves, numerous attempts have been made to extend these results to higher dimensions. Ein and Lazarsfeld proved that if A is a very ample line bundle, then K_X + mA satisfies property N_p for any m>=n+1+p. It has ever since been an open question if the same holds true for A ample and basepoint free. In joint work with Purnaprajna Bangere we give a positive answer to this question.

Elliptic surfaces fibering over an elliptic curve with 12 singular fibers form a ten-dimensional moduli space. Their middle cohomology has a K3 type Hodge structure endowing the moduli space with a period map into a ten-dimensional arithmetic quotient. I will discuss a proof that this period map is dominant (like moduli of K3 surfaces) but that the degree of the period map exceeds one (unlike moduli of K3 surfaces). This is joint work with F. Greer and A. Ward.

In this talk I will study if some well—known properties of log canonical threefold singularities over the complex numbers hold true over perfect fields of positive characteristics? For surfaces of Fano type the Kawamata–Viehweg vanishing theorem, which in general famously fails in positive characteristic p, hold true if p>5. This has the consequence that threefold klt singularities are Cohen–Macaulay in this characteristic. With this positive result at hand it is natural to ask if Kollárs theorem, asserting that the depth of a log canonical three-dimensional singularity is three at any closed point which itself is not a log canonical center, also can be extended to positive (possibly large enough) characteristics? Another natural and related question is if minimal log canonical centers are normal in (possibly large enough) positive characteristics? In this talk we will answer these two questions. This includes joint work with Bernasconi–Patakfalvi and Posva.

A full strong exceptional collection is an important structure to have on a derived category with many useful implications. For instance, such a collection produces a basis for the Grothendieck group and a tilting object. In this talk, we will discuss the existence of full strong exceptional collections consisting of vector bundles on certain linear GIT quotients by a split reductive group G of rank two. These vector bundles will come from irreducible G-representations whose weights lie in a particular “window” in the weight space of G. This is based on joint work with Daniel Halpern-Leistner.