Algebraic Geometry Seminar

In this talk, I will explore various compactifications of moduli spaces of genus six curves, delving into their intricate interactions. A key ingredient is the K-moduli spaces of log Fano pairs. The intricate wall crossing structure not only yields novel compactifications but also plays a pivotal role in understanding the Hassett-Keel program of moduli spaces of curves.

Classification of singularities is an interesting problem in many areas of algebraic geometry, like the minimal model program. One classical approach is to assign to a singular subvariety a rational number, its log canonical threshold. For complex hypersurface singularities, this invariant has been refined by M. Saito to the minimal exponent. This invariant is related to Bernstein-Sato polynomials, Hodge ideals and higher du Bois and higher rational singularities. In joint work with Qianyu Chen, Mircea Mustață and Sebastián Olano, we defined the minimal exponent for LCI subvarieties of smooth complex varieties. We related it to local cohomology, higher du Bois and higher rational singularities. I will describe what was done in the hypersurface case, give our definition in the LCI case and explain the relation to local cohomology modules and the classification of singularities.

A common issue one can run into is that topological algebra doesn't play well with homological algebra: for example, the category of topological abelian groups is not an abelian category. This introduces many pathologies, for example: continuous group cohomology does not give rise to long exact sequences. Clausen and Scholze (resp, Barwick and Haine) suggest a solution to this kind of problem by extending the category of topological abelian groups to the abelian category of condensed abelian groups (resp. pyknotic abelian groups), which are sheaves of abelian groups on the pro-etale site of a geometric point. We will explain a construction of continuous group cohomology in this condensed setting, give comparisons to classical continuous group cohomology, and explain some consequences and future directions that fall out of this formalism (including, if time allows, to p-adic and perfectoid geometry).

A famous conjecture by Iitaka predicts that, if \(f\colon X\to Y\) is a fibration of smooth projective varieties over \(\mathbb{C}\) with general fiber \(F\), then \(\kappa(K_X)\geq \kappa(K_F)+\kappa(K_Y)\). It was recently shown by Chang that a similar inequality holds for the anticanonical divisors: when the stable base locus \(\mathbf{B}(-K_X)\) is vertical, then \(\kappa(-K_X)\leq \kappa(-K_F)+\kappa(-K_Y)\). Both the Iitaka conjecture and Chang's theorem are known to be false over fields of positive characteristic. However the expectation is that these inequalities should still hold whenever \(F\) is sufficiently well-behaved with respect to the Frobenius morphism. In this talk I will explain how to recover Chang's theorem for this kind of tame fibrations in characteristic \(p>0\) and discuss some related questions. This is based on joint work with M. Benozzo and C.K. Chang.

I will discuss how the recent progress in p-adic Hodge theory could be used to prove a version of Lefschetz Hyperplane Theorem for flat cohomology. Then, I will discuss how this result could be used to recover a recent result of Česnavičius and Scholze saying that \(\mathrm{Pic}(X)_{\mathrm{tors}} = 0\) for any (possibly singular) complete intersection \(X\subset \mathbb P^N\) of dimension at least 2.

Fano varieties are one of the three building blocks of algebraic varieties. In this talk, we will discuss how to describe a general rational Fano variety. Although there is no consensus on how to answer to this question, we will explore some new invariants motivated by combinatorics and toric geometry that may lead to a first approximation of an answer.

Using the division polynomials for elliptic curves in Weierstrass form, it is shown that the group of rational points on the curve \(H:k(y^3 −y)= l(x^3 −x)\) is torsion-free.

The list of projective varieties known to have full strong exceptional collections of coherent sheaves is quite short. There is projective space and a few other homogeneous spaces and toric varieties. One example is the toric variety associated to the "permutahedron," which Castravet and Tevelev studied because of its proximity to the moduli space of pointed rational curves. Since the sum of the betti numbers of this toric variety is n!, Castravet-Tevelev find a new way to count to n! that is intriguing, and which Alicia and I have been modifying to work in the B_n case in which the permutahedron is replaced with the signed permutahedron.