arXiv subjects

Xuqiang Qin

Publications and source records attributed to Xuqiang Qin.

9 recordsLinked to original sources

arXiv2026

Birational geometry of Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls

We characterize the birational geometry of some hyperkähler fourfolds of Picard rank $3$ obtained as the Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls. In each of the two cases considered, we identify all of the birational models, relating each model to familiar geometric constructions, and give explicit birational maps between them. We also provide structural results about the birational automorphism groups, giving generators in both cases and a full set of relations in one case. Finally, as a byproduct of our analysis, we obtain non-isomorphic cubic fourfolds whose Fano varieties of lines are birationally equivalent.

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arXiv2023

Bridgeland stability of minimal instanton bundles on Fano threefolds

We prove that minimal instanton bundles on a Fano threefold $X$ of Picard rank one and index two are semistable objects in the Kuznetsov component $\mathsf{Ku}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macrì and Stellari. When the degree of $X$ is at least $3$, we show torsion free generalizations of minimal instantons are also semistable objects. As a result, we describe the moduli space of semistable objects with same numerical classes as minimal instantons in $\mathsf{Ku}(X)$. We also investigate the stability of acyclic extensions of non-minimal instantons.

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arXiv2022

Birational geometry of Beauville-Mukai systems III: asymptotic behavior

Suppose that a Hilbert scheme of points on a K3 surface S of Picard rank one admits a rational Lagrangian fibration. We show that if the degree of the surface is sufficiently large compared to the number of points, then the Hilbert scheme is the unique hyperkähler manifold in its birational class. In particular, the Hilbert scheme is a Lagrangian fibration itself, which we realize as coming from a (twisted) Beauville-Mukai system on a Fourier-Mukai partner of S. We also show that when the degree of the surface is small our method can be used to find all birational models of the Hilbert scheme.

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arXiv2022

Birational geometry of Beauville-Mukai systems I: the rank three and genus two case

We study wall-crossing for the Beauville-Mukai system of rank three on a general genus two K3 surface. We show that such a system is related to the Hilbert scheme of ten points on the surface by a sequence of flops, whose exceptional loci can be described as Brill-Noether loci. We also obtain Brill-Noether type results for sheaves in the Beauville-Mukai system.

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arXiv2022

Birational geometry of Beauville-Mukai systems II: general theory in low ranks

Via wall-crossing, we study the birational geometry of Beauville-Mukai systems on K3 surfaces with Picard rank one. We show that there is a class of walls which are always present in the movable cones of Beauville-Mukai systems. We give a complete description of the birational geometry of rank two Beauville-Mukai systems when the genus of the surface is small.

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arXiv2021

Compactification of the moduli space of minimal instantons on the Fano threefold $V_4$

We study semi-stable sheaves of rank $2$ with Chern class $c_1=0$, $c_2=2$ and $c_3=0$ on the Fano 3-folds $V_4$ of Picard number $1$, degree $4$ and index $2$. We show the moduli space of such sheaves is isomorphic to the moduli space of semi-stable rank $2$ even degree vector bundles on a genus $2$ curve. This provides a natural smooth compatification of the moduli space of Ulrich bundles of rank $2$ on $V_4$.

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arXiv2020

Compactification of the moduli space of instanton sheaves on the Fano threefold $V_5$

We study semistable sheaves of rank $2$ with Chern classes $c_1=0$, $c_2=2$ and $c_3=0$ on the Fano 3-fold $V_5$ of Picard number $1$, degree $5$ and index $2$. We show that the moduli space of such sheaves has a component that is isomorphic to $\mathbb{P}^5$ by identifying it with the moduli space of semistable quiver representations. This provides a natural smooth compactification of the moduli space of minimal instantons, as well as Ulrich bundles of rank $2$ on $V_5$.

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arXiv2019

Moduli of quiver representations for exceptional collections on surfaces

Suppose $S$ is a smooth projective surface over an algebraically closed field $k$, $\mathcal{L}=\{L_1,\ldots,L_n\}$ is a full strong exceptional collection of line bundles on $S$. Let $Q$ be the quiver associated to this collection. One might hope that $S$ is the moduli space of representations of $Q$ with dimension vector $(1,\ldots,1)$ for a suitably chosen stability condition $θ$: $S\cong M_θ$. In this paper, we show that this is the case for del Pezzo surfaces. Furthermore, we show the blow-up at a point can be recovered from an augmentation of exceptional collections (in the sense of L. Hille and M.Perling) via morphism between moduli of quiver representations.

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arXiv2018

Blow ups of $\mathbb{P^n}$ as quiver moduli for exceptional collections

Suppose $P^n_m$ is the blow up of $\mathbb{P}^n$ at a linear subspace of dimension $m$, $\mathcal{L}=\{L_1,\ldots,L_r\}$ is a (not necessarily full) strong exceptional collection of line bundles on $P^n_m$. Let $Q$ be the quiver associated to this collection. One might wonder when is $P^n_m$ the moduli space of representations of $Q$ with dimension vector $(1,\ldots,1)$ for a suitably chosen stability condition $θ$: $S\cong M_θ$. In this paper, we achieve such isomorphism using $\mathcal{L}$ of length $3$. As a result, $P^n_m$ is the moduli space of representations of a very simple quiver. Moreover, we realize the blow up as morphism of moduli spaces for the same quiver.

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