We study mutually one-point incident 22 planes in a smooth cubic 4-fold in char. 2, and cubic 4-folds with 11 cusps in char. 3. (1) Such 22 planes (resp. 11 cusps) appear for every cubic 4-fold associated with supersingular K3 surface in char. 2 (resp. in char. 3 with Artin invariant at most 6). (2) They are Fermat and Klein type and have an action of the Mathieu groups M22 and M11, respectively, in the most special case of Artin invarinat 1.

We prove the Kuznetsov components of a series of hypersurface in projective space reconstruct the hypersurfaces. Our method allows us to work for hypersurfaces of weighted projective space, and obtain the reconstruction theorem of veronese double cone, which is a long-time opening case. I will show how to construct infinitesimal variation of Hodge structure from certain Kuznetsov components. Using classical generic Torelli theorem, this implies Kuznetsov components reconstruct the algebraic variety generically. Joint with J. Rennemo and ShiZhuo Zhang.

Deuring gave a now classical formula for the number of supersingular elliptic curves in characteristic \(p\). We generalize this to a formula for the cycle class of the supersingular locus in the moduli space of principally polarized abelian varieties of given dimension \(g\) in chacteristic \(p\). The formula determines the class up to a multiple and shows that it lies in the tautological ring. We also give the multiple for \(g\) up to \(4\). This is joint work with S. Harashita.

Moduli spaces of stable sheaves on Fano threefolds are known to exhibit pathological behavior in general. Meanwhile, for certain specific cases—such as ideal sheaves of curves with small degree and genus in the cubic threefold, or moduli spaces of lower-rank aCM bundles—these spaces are well-behaved.
From a modern derived categorical perspective, we have the so-called Kuznetsov component \(\text{Ku}(X)\) in \(D(X)\). The well-behaved moduli spaces mentioned above actually parametrize stable objects within \(\text{Ku}(X)\). In this talk, I will begin by recapping this framework with a detailed overview of known results. I will then present our recent work on higher-dimensional moduli spaces of stable objects in \(\text{Ku}(X)\).
This is a joint work with Yingbang Lin, Laura Pertusi, and Xiaolei Zhao.

There are two basic objects in projective algebraic geometry: one is a variety of minimal degree and the other is a del Pezzo variety. In this talk, I'd like to introduce higher secant varieties of minimal degree and del Pezzo higher secant varieties to nonexpert with modest backgrounds. Recently, classification and characterization of such varieties have been paid attention.  I will also give many interesting examples explaining main results.