For a strictly ample line bundle L on a curve C, a minimal algebra generating set for the section ring of L gives an embedding of C into a weighted projective space. We will discuss the syzygies of these embeddings over their corresponding nonstandard graded polynomial rings for line bundles of the form O(dP), where P is a point on C.
Associated to any regular matroid of rank g on k elements, one can associate a multivariable semistable degeneration of principally polarized abelian g-folds over a k-dimensional base. I will discuss joint work with de Gaay Fortman and Schreieder, proving that a combinatorial invariant of the matroid obstructs the algebraicity of the minimal curve class, on the very general fiber of the associated degeneration. Corollaries include the failure of the integral Hodge conjecture for abelian varieties of dimension ≥ 4 and the stable irrationality of very general cubic threefolds.
The Enriques criterion for unirationality of conic bundles states that if X is a conic bundle over P^2, then X is unirational if and only if the conic bundle admits a rational multisection. Furthermore, if X is unirational but not rational, then such a rational multisection necessarily has even degree. In this talk, we study non-existence of degree 2 rational multisections. This is joint work with Jeffrey Diller and Eric Riedl.
I will show that any prime Fano threefold X of genus 8 is a linear section of the 8-dimensional Grassmann variety Gr(6,2). This generalizes a result of Gushel and Mukai in characteristic zero. As applications, I will also show that (1) X is birational to a cubic threefold and hence is irrational, and (2) X is F-split if the characteristic is greater than 2. (joint work with Hiromu Tanaka)
In this talk, I will discuss Grauert–Riemenschneider vanishing and Steenbrink vanishing for three-dimensional F-pure singularities. I will also explain applications to the logarithmic extension theorem for one-forms.
We discuss some recent and forthcoming work on finitely presented algebras over valuation rings. Specifically, we will discuss properties around regularity and dualizing complexes. We will also explain how Noetherian approximation is used to generalize known results for Noetherian rings to our setup. Time permits, we will give applications of our result back to Noetherian rings.
When R is a non-smooth k-algebra, its ring of algebraic differential operators is in practice complicated to pin down and does not appear to have the same well-behaved features as in the smooth setting. To remedy this, in recent decades a new approach developing derived different operators has been developed in the work of Jeffries and of Smith and Van den Bergh. More recently, Yang developed a differential graded (dg) algebra model for these, as did Jiang, thus giving a desired algebra structure to these. In joint work with Eamon Gallego-Quinlan, Jack Jeffries, Devlin Mallory, and Josh Pollitz, we have developed several alternative definitions/models for derived differential operators some of which we have found are more natural for applications and some more concrete for computations, and whose cohomology and resulting algebra structures agree with theirs. We will introduce these and, if time permits, focus particularly on a derived analog of a classical interpretation of the ordinary ring of differential operators in terms of the Frobenius endomorphism.
In this talk, we study the precise inversion of adjunction (PIA) conjecture and the lower semi-continuity (LSC) conjecture for hyperquotient singularities. Previously known results for these conjectures in this setting required the assumption that the singularity is klt, and without this assumption, a counterexample to the PIA conjecture was known to exist. To overcome this limitation, we introduce a localized notion of virtually free actions and characterize it in terms of the arc spaces of quotient varieties. Utilizing this characterization, we establish a necessary and sufficient condition for the PIA conjecture to hold for arbitrary hyperquotient singularities. Furthermore, as an application of this analysis, we unconditionally prove that the LSC conjecture holds for hyperquotient singularities, even if they are not klt. This is a joint work with Kohsuke Shibata.
The moduli space of smooth hypersurfaces in projective space can be constructed as a GIT quotient by linear changes of coordinates, and it comes with a natural GIT compactification. In certain degrees and dimensions, Hodge theory provides a second compactification via the period map, namely the Baily-Borel compactification. Building on recent progress on higher singularities and a new stability criterion formulated in terms of the minimal exponent (a refinement of the log canonical threshold), I will discuss the birational geometry of these two compactifications and describe consequences for the boundary behavior of the period map.
A fundamental problem at the confluence of algebraic geometry, commutative algebra and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on (partial) flag varieties. Over fields of characteristic zero, this is the content of the Borel-Weil-Bott theorem and is well-understood, but in positive characteristic it remains wide open, despite important progress over the years. In my talk I will describe recent developments in the case of the incidence correspondence - the partial flag variety consisting of pairs of a point in projective space and a hyperplane containing it. Joint work with Annet Kyomuhangi, Emanuela Marangone, and Ethan Reed.
In 1998, Hara established a complete classification of two-dimensional surface singularities that are either strongly F-regular or F-pure. This classification revealed that every klt surface singularity in characteristic p > 5 is strongly F-regular, whereas the property of being F-pure depends sensitively on the characteristic. In this talk, we focus on the notion of quasi-F-splitting, recently introduced by Yobuko as a weaker version of F-purity, and we show that every two-dimensional log canonical singularity in characteristic p > 3 is quasi-F-splitting.
In this talk, I start with a brief review on the basic properties of perfectoid towers. Then I will explain its construction and applications to singularities in mixed characteristic. This is a report on work partly obtained with R. Ishizuka, S. Ishiro and K. Hayashi.
This is a report on joint work with D. Bath and A. De Valle Rodriguez.
The Grothendieck—Deligne comparison asserts an equality between topology and algebra via the algebraic de Rham complex. Motivated by work of Deligne and Brieskorn, K. Saito developed the theory of "logarithmic differentials". A long-standing question is to determine when the "logarithmic de Rham complex" is equivalent to the full one. In the talk we first discuss the known cases. The commonality between these is a homogeneity condition on the divisor. We explain a 2002 conjecture on this homogeneity phenomenon by Calder\'on Moreno, Mond, Narv\'aez Macarro and Castro Jim\'enez. We generalize it by removing a rather stringent hypothesis on the divisor, and prove it in dimension 3.
I will begin by reviewing the theory of higher singularities in characteristic zero, and then move on to higher singularities in positive characteristic. Finally, I will discuss some recent developments in the theory.
The plus-pure threshold is a mixed-characteristic analogue of the F-pure threshold and is closely related to perfectoid purity and BCM-regularity. We establish lower bounds for plus-pure thresholds of diagonal hypersurfaces, analogous to Hernández's result on F-pure thresholds.
Quasi-F-splitting is a generalization of F-splitting and if a scheme is quasi-F-split, then it admits a lifting modulo p^2. In this talk, I will discuss deformation-theoretic aspects of such liftings. This is ongoing work.
In positive characteristic, Frobenius splitting provides powerful tools for studying vanishing theorems and singularities. In this talk, I will discuss mixed-characteristic analogues of this notion for smooth hypersurfaces. A quasi-compact separated scheme is called perfectoid split if it admits a perfectoid cover such that the natural map on structure sheaves admits a splitting. This notion may be viewed as a mixed-characteristic analogue of Frobenius splitting. Moreover, Bhatt proved that regular perfectoid split projective schemes satisfy a Kodaira-type vanishing theorem.
The main result of the talk is that smooth Calabi–Yau hypersurfaces of degree d are perfectoid split, provided that p is larger than the relative dimension and p does not divide d. I will also explain how the proof is reduced to uniform bounds for splitting-order sequences and their relationship with perfectoid purity.