Abstracts and Schedule

Schedule

Saturday:

• 9:00-9:30: Breakfast

• 9:30-10:30: Talk 1-Alessio Cela

• 10:30-10:45: Break

• 10:45-11:45: Talk 2-Erin Dawson

• 11:45-1:00: Lunch

• 1:00-2:00: Talk 3-Roya Beheshti

• 2:00-2:30: Coffee Break

• 2:30-3:30: Talk 4-Eric Riedl

• 3:30-3:45: Break

• 3:45-4:45: Talk 5-Carl Lian

Sunday:

• 8:30-9:00: Breakfast

• 9:00-10:00: Talk 6-Erin Dawson

• 10:00-10:15: Break

• 10:15-11:15: Talk 7-Alessio Cela

• 11:15-11:45: Coffee Break

• 11:45-12:45: Talk 8-Carl Lian

Abstracts

• Talk 1-Introduction to Fixed domain curve counts (Alessio Cela)

Abstract: In this talk, I will introduce the notion of geometric and virtual Tevelev degrees, explain the differences between them, and discuss some cases of agreement. I will then describe the computation of the virtual degrees for projective spaces, following the work of Bertram-Daskalopoulos-Wentworth and Buch-Pandharipande. 

• Talk 2 & Talk 6-Tropical Tevelev degrees of P^1 (Erin Dawson)

Abstract: Tropical Hurwitz spaces parameterize genus g, degree d covers of the tropical line with fixed branch profiles. Since tropical curves are metric graphs, this gives us a combinatorial way to study Hurwitz spaces. Tevelev degrees are the degrees of a natural finite map from the Hurwitz space to a product M_{0,n} cross M_{g,n}. In 2021, Cela, Pandharipande and Schmitt presented this interpretation of Tevelev degrees in terms of moduli spaces of Hurwitz covers. In these talks I will show that these tropical enumerative invariants agree with their algebraic counterparts, as well as explore a method to perform this calculation of Tevelev degrees of P^1 using the moduli spaces of tropical Hurwitz covers.

• Talk 3-Asymptotic Enumerativity of Tevelev degrees (Roya Beheshti)

Abstract: The question of asymptotic enumerativity of Tevelev degrees for Fano varieties was proposed by Lian and Pandharipande and proven by them for smooth hypersurfaces of low degree. After reviewing their work, I will report on joint work with Lehmann, Lian, Riedl, Starr, and Tanimoto, where we improve the Lian-Pandharipande bound and provide counterexamples to asymptotic enumerativity for certain other Fano varieties.

• Talk 4-Stability of vector bundles and jumping loci (Eric Riedl)

Abstract: If a curve in a smooth variety X passes through the expected number of general points, this implies that the tangent bundle T_X has to have a special property, called semi-stability. This talk will be an introduction to the notion of stability of vector bundles, and will talk about what happens when a semi-stable vector bundle is restricted to a general curve. We will also talk about bounds on the dimension of the jumping locus, where the restricted tangent bundle is less stable than predicted. This is joint work with Brian Lehmann and Sho Tanimoto (and some with Anand Patel and Dennis Tseng).

• Talk 5 & Talk 8-Tevelev degrees of P^r  (Carl Lian)

Abstract: In how many ways can a general curve (C,p_1,...,p_n) of genus g be threaded through n general points in projective space? Castelnuovo determined the answer via Schubert calculus when n is as small as possible in 1889. At the other extreme, when n is large, the much simpler answer (r+1)^g is determined by the Vafa-Intriligator formula, re-proven many times since the 1990s. I will explain how to interpolate between these two very different answers. The key engine is the moduli space of complete collineations, which allows for a sequence of degenerations making contact with both of the earlier calculations. Time-permitting, I will explain how the new methods may be indicating where to go from here.

• Talk 7-Tevelev Degrees of Blow-ups of Projective Spaces (Alessio Cela) 

Abstract: I will explain how to compute the Tevelev degrees of point blow-ups of projective spaces. In genus 0 and higher genus for large degrees, the geometric and virtual Gromov-Witten counts agree in Fano (and some (-K)-nef) examples, though not in general. For toric blow-ups, geometric counts can be expressed as integrals over products of Jacobians and symmetric products of the domain curve, and can be explicitly evaluated in genus 0 and for blow-ups of P^r at a point. Time permitting, I will also explain how to handle low (anticanonical) degree counts.