代数几何

We study automorphism groups of del Pezzo surfaces without points over a field of zero characteristic, and estimate their Jordan constants.

Synthetic algebraic geometry uses homotopy type theory extended with three axioms to develop algebraic geometry internal to a higher version of the Zariski topos. In this article we make no essential use of the higher structure and use…

Let $X$ be a smooth variety over a field of characteristic $p$. It is a natural question whether the Frobenius pushforwards $F_*^e\mathcal O_X$ of the structure sheaf are tilting bundles. We show if $X$ is a smooth del Pezzo surface of…

In arXiv:2404.19088, we initiated a program linking birational invariants with smooth ones and offering new interpretations of classical invariants, such as the Kervaire-Milnor invariants. Here, we rely on the profound geometric reasoning…

First we refine the duality theory for Witt divisorial sheaves on smooth projective varieties over a perfect field of positive characteristic. Building on previous work [Lem22], we remove the residual derived limit to obtain a cleaner…

We determine the generic consistency, dimension and nondegeneracy of the zero locus over $\mathbb{C}^*$, $\mathbb{R}^*$ and $\mathbb{R}_{>0}$ of vertically parametrized systems: parametric polynomial systems consisting of linear…

Let $\mathcal{M}_{g,n}$ be the moduli space of algebraic curves of genus $g$ with $m+n$ marked points decomposed into the disjoint union of two sets of cardinalities $m$ and $n$, and $H_c^{\bullet}(\mathcal{M}_{m+n})$ its compactly…

We extend the mirror construction of singular Calabi-Yau double covers, introduced by Hosono, Lee, Lian, and Yau, to a broader class of singular Calabi-Yau $(\mathbb{Z}/2)^k$-Galois covers, and prove Hodge number duality for both the…

For a very general polarized $K3$ surface $S\subset \mathbb{P}^g$ of genus $g\ge 5$, we study the linear system on the Hilbert square $S^{[2]}$ parametrizing quadrics in $\mathbb{P}^g$ that contain $S$. We prove its very ampleness for…

We discuss the holonomic dual of tautological systems, with a view towards applications to linear free divisors and to homogeneous spaces. As a technical tool, we consider a Chevalley--Eilenberg type complex, generalizing Euler--Koszul…

In the present paper we construct quadratic equations and linear syzygies for tangent varieties using 4-way tensors of linear forms and generalize this method to higher secant varieties of higher osculating varieties. Such equations extend…

Consider the strata of primitive $k$-differentials on the Riemann sphere whose singularities, except for two, are poles of order divisible by $k$. The map that assigns to each $k$-differential the $k$-residues at these poles is a ramified…

We construct a perfect version of Morel--Voevodsky's motivic homotopy category over a perfect base scheme in positive characteristic. By checking the axioms of a coefficient system, we establish a six-functor formalism. We show that…

We investigate the toric geometry of two families of generalised determinantal varieties arising from permutations: Matrix Schubert varieties ($\overline{X_w}$) and Kazhdan-Lusztig varieties ($\mathcal{N}_{v,w}$). Matrix Schubert varieties…

In this paper, we study the vertex functions of finite type A bow varieties. Vertex functions are K-theoretic analogs of I-functions, and 3d mirror symmetry predicts that the q-difference equations satisfied by the vertex functions of a…

Tropical caustic of a convex domain on the plane is a canonically associated tropical analytic curve inside the domain. In this note we give a graphical proof for the classification of its intermediate vertices, implying in particular that…

Given a K3^[2]-type manifold X with a symplectic involution i, the quotient X/i admits a Nikulin orbifold Y as terminalization. We study the symplectic action of a group G of order 4 on X, such that i belongs to G, and the natural…

We prove that the transcendental Brauer group of a K3 surface X over a finitely generated field k is finite, unless k has positive characteristic p and X is supersingular, in which case it is annihilated by p.

We prove that the quotient of the Brauer group of a product of varieties over k by the sum of the images of the Brauer groups of factors has finite exponent. The bulk of the proof concerns p-primary torsion in characteristic p. Our approach…

Call a Laurent polynomial $W$ `complete' if its Newton polytope is full-dimensional with zero in its interior. We show that if $W$ is any complete Laurent polynomial with coefficients in the positive part of the field $K$ of generalised…