arXiv subjects
Publications and source records attributed to Huachen Chen.
Using the algebraic criterion proved by Bandiera, Manetti and Meazzini, we show the formality conjecture for universally gluable objects with linearly reductive automorphism groups in the bounded derived category of a K3 surface. As an application, we prove the formality conjecture for polystable objects in the Kuznetsov components of Gushel--Mukai threefolds and quartic double solids.
We show that Brill--Noether loci in Hilbert scheme of points on a smooth connected surface $S$ are non-empty whenever their expected dimension is positive, and that they are irreducible and have expected dimensions. More precisely, we consider the loci of pairs $(I, s)$ where $I$ is an ideal that locally at the point $s$ of $S$ needs a given number of generators. We give two proofs. The first uses Iarrobino's descriptionof the Hilbert--Samuel stratification of local punctual Hilbert schemes, and the second is based on induction via birational relationships between different Brill--Noether loci given by nested Hilbert schemes.
We prove that given a cubic fourfold $Y$ not containing any plane, the Voisin map $v: F(Y)\times F(Y) \dashrightarrow Z(Y)$ constructed in \cite{Voi}, where $F(Y)$ is the variety of lines and $Z(Y)$ is the Lehn-Lehn-Sorger-van Straten eightfold, can be resolved by blowing up the incident locus $Γ\subset F(Y)\times F(Y)$ endowed with the reduced scheme structure. Moreover, if $Y$ is very general, then this blowup is a relative Quot scheme over $Z(Y)$ parametrizing quotients in a heart of a Kuznetsov component of $Y.$
We observe that O'Grady's birational maps between moduli of sheaves on an elliptic K3 surface can be interpreted as intermediate wall-crossing (wall-hitting) transformations at the so-called totally semistable walls, studied by Bayer and Macrì. As an ingredient to prove this observation, we describe the first totally semistable wall for ideal sheaves of $n$ points on the elliptic K3. We then use this observation to make a remark on Marian and Oprea's strange duality.