Algebraic Geometry Seminar

What allowed for many developments in algebraic geometry (especially birational geometry) and commutative algebra was a discovery of the notion of a Frobenius splitting. Recently, Yobuko introduced a more general concept, a quasi-F-splitting, which captures more refined arithmetic invariants related to Crystalline Cohomology. In my talk, I will discuss on-going projects in which we continue the development of the theory of quasi-F-splittings. This is based on joint work with Tatsuro Kawakami, Hiromu Tanaka, Teppei Takamatsu, Fuetaro Yobuko, and Shou Yoshikawa.

Stable varieties are higher dimensional analogues of stable curves. In characteristic zero, it is known that their moduli space is a projective variety. However, this is not known yet in positive characteristic. In this talk, I will first survey recent results about stable surfaces in positive characteristic due to Arvidsson, Bernasconi, Patakfalvi and myself. Then I will sketch an ongoing approach to the properness of the corresponding moduli functor.

Let \(R \rightarrow S\) be a pure map of rings. If \(S\) is regular, then it is known that \(R\) has rational singularities, and hence is Cohen-Macaulay. This result applies in particular to inclusions of rings of invariants by linearly reductive groups, split maps, and faithfully flat maps.
In this talk, I will discuss my recent work (joint with Charles Godfrey) showing that if \(R\) and \(S\) are essentially of finite type over the complex numbers, and \(S\) has Du Bois singularities, then \(R\) has Du Bois singularities. Our result is new even when \(R \rightarrow S\) is faithfully flat. As a consequence, under the same hypotheses on \(R \rightarrow S\), if \(S\) has log canonical type singularities and the canonical divisor on \(R\) is Cartier, then \(R\) has log canonical singularities.

We discuss the Mori dream space (MDS) property for blow-ups of toric surfaces defined by rational plane triangles at a general point. We consider a parameter space of all such triangles and show how it can be used to prove the MDS property for these varieties. Furthermore, this parameter space helps explain most known results in the area and has also led to surprising new results, including examples of such surfaces with a semi-open effective cone. This is joint work with José Luis González and Kalle Karu.

I will talk about moduli spaces of rational curves on Fano hypersurfaces and discuss questions regarding when these moduli spaces are uniruled or do not contain many rational curves. One motivation for the study of these types of questions comes from the question of when Fano hypersurfaces are swept out by certain families of rational surfaces. This is based on joint work with Eric Riedl.

For a normal variety \(X\), we say \(X\) satisfies the extension theorem if, for every proper birational morphism from \(Y\), every differential form on the regular locus of \(X\) extends to \(Y\). This is a basic property relating differential forms and singularities, and many results are known over the field of complex numbers. In this talk, we discuss the extension theorem in positive characteristic. Existing studies depend on geometric tools such as log resolutions, (mixed) Hodge theory, the minimal model program, and vanishing theorems, which are not expected to be true or are not known for higher-dimensional varieties in positive characteristic. For this reason, I introduce a new algebraic approach to the extension theorem using Cartier operators. I also talk about an application of the theory of quasi-F-splitting, which is studied in joint work with Takamatsu-Tanaka-Witaszek-Yobuko-Yoshikawa, to the extension problem.

There is a beautiful combinatorial and geometric story connecting a polytope known as the permutohedron, the algebra of the symmetric group, and the geometry of a particular moduli space of curves first studied by Losev and Manin. I will describe these three worlds and their connection to one another, and then I will discuss joint work with C. Damiolini, D. Huang, S. Li, and R. Ramadas that generalizes the story by introducing a new family of polytopal complexes and relating it to a new family of moduli spaces.

In this talk we will consider the formal-local singularity type of points on an algebraic variety X. Extending work of Ephraim and Hauser-Mueller, we show that the subset of points with a fixed singularity type is locally closed in the Zariski topology; and the associated reduced subscheme, called isosingular locus, is smooth. Moreover, over a field of characteristic 0, we prove the existence of a decomposition of the formal neighborhood at each point. In positive characteristics we give counterexamples and relate them to questions on finite determinacy. Finally, we discuss connections to the local geometry of arc spaces and in particular the Drinfeld-Grinberg-Kazhdan theorem. This talk covers joint work with Herwig Hauser.

We look at a family over the punctured disk of polarized abelian varieties endowed with an automorphism and to ask whether we can complete the central fiber so that the map remains an automorphism.

Let \((Y, D)\) be a log Calabi-Yau threefold, meaning that \(Y\) is a smooth projective threefold over \(\mathbb{C}\) and \(D \subset Y\) is a normal crossing divisor such that \(K_{Y}+D\) is trivial. Moreover, suppose that \(D\) is maximal, meaning there exists a \(0\)-stratum of \(D\). Suppose there exists a \(K3\)-fibration \(f: (Y, D) \rightarrow (\mathbb{P}^{1}, \infty)\) with \(D = f^{\ast}(\infty)\) and \(H^{3}(Y)=0\). Such fibrations arise as mirrors to Fano threefolds. I show that the pseudoautomorphism group of \(Y\) acts on the codimension one faces of the movable effective cone of \(Y\) with finitely many orbits. This is implied by the Kawamata-Morrison-Totaro cone conjecture.

In this talk we will discuss higher Fano manifolds, which are Fano manifolds with positive higher Chern characters. In particular we will focus on toric 2-Fano manifolds. Motivated by the problem of classifying toric 2-Fano manifolds, we will introduce a new invariant for smooth projective toric varieties, the minimal projective bundle dimension, \(m(X)\). This invariant \(m(X)\) captures the minimal degree of a dominating family of rational curves on \(X\) or, equivalently, the minimal length of a centrally symmetric primitive relation for the fan of \(X\). We'll present a classification of smooth projective toric varieties with \(m(X) ≥ dim(X)-2\), and show that projective spaces are the only 2-Fano manifolds among smooth projective toric varieties with \(m(X)\) equal to 1, \(dim(X)-2\), \(dim(X)-1\), or \(dim(X)\). This is joint work with Carolina Araujo, Roya Beheshti, Ana-Maria Castravet, Svetlana Makarova, Enrica Mazzon, and Nivedita Viswanathan.

This talk is an invitation to the study of monodromy conjecture for hypersurfaces in complex affine spaces. I will recall two different ways to understand singularities of hypersurfaces in complex affine spaces. The first one is to use D-modules to define the b-function (also known as the Bernstein-Sato polynomial) of a polynomial (defining the hypersurface). The other one uses motivic integrals and resolution of singularities to obtain the motivic/topological zeta function of the hypersurface. The monodromy conjecture predicts that these two ways of understanding hypersurface singularities are related. Then I will discuss some known cases of the conjecture for hyperplane arrangements. There will be plenty of examples.