Convegno
“A TOUR THROUGH ALGEBRAIC GEOMETRY”

Title: Log/local quasi-map correspondence Abstract: I will explain an equivalence between the virtual count of rational curves in the total space of an anti-nef line bundle and the virtual count of rational curves maximally tangent to a smooth section of the dual line bundle due to Graber--van Garrel--Ruddat. I will present generalisations which involve quasi-maps. This is work with A. Cobos-Rabano and Q Shafi.

Derived enumerative geometry Abstract: Derived algebraic geometry provides a powerful set of tools to enumerative geometers, giving geometric spaces which encode the "virtual structures" of the moduli problems . I will discuss a joint work with D. Karn, E. Mann and C. Manolache in which we define a derived enhancement for the moduli space of sections.

Title: Terminal 3-folds that are not Cohen-Macaulay Abstract: An important local vanishing theorem for the minimal model program is the fact that klt singularities in characteristic zero are Cohen-Macaulay. In contrast, even in the narrow setting of terminal singularities of dimension 3, we show that Cohen-Macaulayness can fail in characteristic p or mixed characteristic (0,p) for p equal to 2, 3, or 5. This is optimal, by work of Arvidsson-Bernasconi-Lacini. (These examples help to explain why the MMP remains an open question for 3-folds in characteristic 2 or 3.) The examples are quotients of regular schemes by the cyclic group G of order p. In characteristic p or mixed characteristic (0,p), such quotients can exhibit a wide range of behavior.

Title: Tableaux Littlewood—Richardson rules for 2-step flags Abstract: The Abelian/non-Abelian correspondence gives rise to a natural basis for the cohomology of flag varieties, which - except for Grassmannians - is distinct from the Schubert basis. I will describe this basis and its multiplication rules, and explain how to relate it to the Schubert basis for two-step flag varieties. I will then explain how this leads to new tableaux Littlewood--Richardson rules for many products of Schubert classes. This is joint work (separately) with Wei Gu and Linda Chen.

Title: The Noether-Lefschetz locus in families of threefolds Abstract: For a fixed threefold X and very ample line bundle H, the Noether-Lefschetz locus parametrizes surfaces in X which are linearly equivalent to H and have Picard rank greater than that of X. We discuss the behavior of special components of the Noether-Lefschetz locus as we deform the pair (X, H), particularly when X is a singular Fano threefold. This is joint work with A. Grassi and J. Rana.