Mahrud Sayrafi: "Splitting of vector bundles on toric varieties"

Abstract: 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. Generalizations of this criterion, under additional hypotheses, have been proven for other toric varieties, for instance by Eisenbud-Erman-Schreyer for products of projective spaces, by Schreyer for Segre-Veronese varieties, and Ottaviani for Grassmannians and quadrics. This talk is about a splitting criterion for arbitrary smooth projective toric varieties, as well as an algorithm for finding indecomposable summands of sheaves and modules in the more general setting of Mori dream spaces.

Feiyang Lin: "Finding special line bundles on special tetragonal curves"

Abstract: There is a canonical way to associate to a degree 4 cover of P^1 two vector bundles E and F, which give rise to a stratification of the Hurwitz space H_{4,g}. It is natural to ask whether the Brill-Noether theory of tetragonal curves is controlled by this data. I will describe a procedure for producing a particular line bundle on tetragonal covers in special strata, which is expected to be special in the Hurwitz-Brill-Noether sense. The main technique is the realization of an inflation of vector bundles on P^1 as a blow-up and blow-down of the associated projective bundle.

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A numerical semigroup S is a cofinite subset of the non-negative integers, containing 0 and closed under addition. They arise as value semigroups of 1-dimensional singularities, as Weierstrass semigroups of points on smooth curves, and the associated semigroup rings form a pleasantly simple family of examples of 1-dimensional domains. 

The smallest nonzero element is called the multiplicity, m(S). Kunz showed that the numerical semigroups of multiplicity m can be represented as the lattice points in a convex rational cone in QQ^(m-1), now called the Kunz cone; and that many properties of the semigroup ring are determined by the face of the Kunz cone on which the semigroup lies.

I'll describe the Kunz cone and some of the still-open problems about semigroup rings that might be studied using it.

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In this talk, we delve into the intricate interplay between vanishing orders of functions along singularities and ideal containment problems in Noetherian rings. We start by focusing on the polynomial ring R = C[x1, . . . , xn] and explore the behavior of prime ideals and their symbolic powers. If P is a prime ideal then the nth symbolic power P(n) is the ideal of polynomial functions that vanish to order at least n at the generic point of V (P ). The Zariski-Nagata Theorem, established by Zariski in 1949, reflects the non- singular nature of n-space by asserting that if P ⊆ Q are prime ideals, then P (n) ⊆ Q(n) for all natural numbers n. However, when dealing with rings with singularities, it is no longer the case that P (n) ⊆ Q(n) for all prime ideal containments P ⊆ Q.

Pioneered by Huneke, Katz, and Validashti, The Uniform Chevalley Theorem introduces a linear adjust- ment, requiring functions to vanish along the generic point of V (P ) to ensure a vanishing order of at least n along the generic point of V (Q). Specifically, it establishes the containment of ideals as P(Cn) ⊆ Q(n) for all P ⊆ Q and n ∈ N, where C depends on the singular prime Q.

A notable limitation of the theorem is the dependency of the constant C on the singular prime Q, which restricts its applicability. This talk will explore methodologies to determine a constant C that remains inde- pendent of the prime Q, thereby enhancing the theorem’s utility. The investigation will also shed light on the role of the Izumi-Rees Theorem and unveil novel uniform properties exhibited by affine rings.

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