代数几何

Let $\calP$ be a general pencil of curves of degree $d$ in the projective plane. In this paper we review the computation of the number of curves in $\calP$ that have a hyperflex line, a flex bitangent line or a tritangent line. Then we…

We prove that the derived category of a branched double cover is equivalent to a category of matrix factorizations for a fiberwise quadratic potential on the associated line bundle. This requires the linear fiber coordinate to have odd…

We show that the completed volumes introduced by Duriev-Goujard-Yakovlev as an approximation to compute Masur-Veech volumes via Witten-Kontsevich's combinatorial classes agrees with the top intersection of the tautological class on the…

In this article, we prove the strong monodromy conjecture for complex hyperplane arrangements by proving a conjecture of Budur, Musta\c t\u a and Teitler that $-n/d$ is a root of the $b$-function of an irreducible essential and central…

In this work, we develop a new theory of multivariate V-filtration on D-modules along a simple normal crossing divisor and relate it with Sabbah's multi-filtration. We establish several new structural results and relate them with the Hodge…

For a classical simple and simply connected group $G$, let $\mathcal{M}_{G,\omega}$ be the moduli space of $\omega$-semistable parabolic $G$-bundles on a complex smooth projective curve of genus $g$. We prove two results in this article:…

We compute the small quantum cohomology of Gushel-Mukai fourfolds. Following [13], our computations imply that the very general ones are not rational. Following [8], and thanks to a suitable deformation of the small quantum cohomology ring,…

Let S be a connected scheme smooth and of finite type over the field of complex numbers. To every 1-motive over S, Andr\'e associated the enriched Hodge realization given by a torsion-free, graded-polarizable and admissible variation of…

On smooth projective varieties of dimension $d$, $d$-tilting bundles are important in both geometry and representation theory, since they provide a bridge from the geometry of such varieties to the derived McKay correspondence and to higher…

We show that the mod $p$ fiber of the Shimurian stack $BT_n^{G,\mu}$ constructed by Gardner--Madapusi is a gerbe over the corresponding stack of truncated displays. This confirms a conjecture of Drinfeld.

We study generalized $V$-filtrations, defined by Sabbah, on $\mathcal D$-modules underlying mixed Hodge modules on $X\times \mathbf A^r$. Using cyclic covers, we compare these filtrations to the usual $V$-filtration, which is better…

Given a split simply connected and connected algebraic group scheme $\mathbb G$ over $\mathbb Z$ and a split parabolic subgroup scheme $\mathbb P\subset \mathbb G$, this paper constructs semi-orthogonal decompositions of the bounded derived…

We show by a uniform argument that every index one prime Fano threefold $X$ of genus $g\geq 6$ can be reconstructed as a Brill-Noether locus inside a Bridgeland moduli space of stable objects in the Kuznetsov component $\mathcal{K}u(X)$. As…

In this paper, we study the twisted Fourier-Mukai partners of abelian surfaces. Following the work of Huybrechts [doi:10.4171/CMH/465], we introduce the twisted derived equivalence between abelian surfaces. We show that there is a twisted…

In this article we provide a classification and description of compact Riemann surfaces admitting a triangular action of a group of order $2p^2,$ where $p$ is an odd prime number. We obtain that all such Riemann surfaces are isomorphic to…

We give a new proof of vanishing result of Esnault for the cohomology of constructible sheaves in the tower of ``mock'' Frobenius covers of projective space. The key idea is to use (a global form of) the perversity of nearby cycles.

The goal of this paper is to describe the birational geometry of the blowup of $\mathbb{P}^n$ at $n+4$ points in very general position. To achieve this, we follow an idea of Mukai and explore a special instance of Gale duality, namely, a…

Absolutely indecomposable vector bundle and parabolic vector bundles are well-studied via quiver representations. In this paper, we study absolutely indecomposable quasi-parabolic $G$-bundles over $\mathbb{P}^1$ with generic additive…

Due to a result by Andreotti and Frankel \cite{andreotti1959}, it can be seen that the complement of a complex projective curve has the homotopy type of a $2$-dimensional CW complex. However, no general method has been given to compute…

The purposes of this article are threefold. First, to determine numerically when an arbitrary blowup of a smooth surface is smooth. We show the surface is smooth if and only if certain rational parameters involving log discrepancy and…