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Qixiao Ma

Publications and source records attributed to Qixiao Ma.

8 recordsLinked to original sources
arXiv2026

Torsors under the universal Jacobian over $\mathcal{M}_g$

We consider the universal family of smooth genus-$g$ curves over $\mathbb{C}$. We show that for $g\geq4$, every torsor under the relative Jacobian is isomorphic to a connected component of the relative Picard scheme. As a byproduct, we show that $\mathrm{Br}(\mathcal{M}_{3,1})=\mathbb{Z}/2\mathbb{Z}$ over $\mathbb{C}$ and $\mathrm{H}_2(\Gamma_{3,1},\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}$, pinning down the torsion subgroup in \cite[Theorem 1.2]{zbMATH01991000}.

math.AG↗
arXiv2024

Torsors of the Jacobians of the universal Fermat curves

Let $m\geq3$ be an integer. We show that every torsor of the Jacobian of the universal family of degree-$m$ Fermat curve is necessarily a connected component of the Picard scheme. We show that the Jacobian of the generic degree-$m$ Fermat curve has uncountably many non-isomorphic torsors. We give some results towards the Franchetta type problem for torsors of the Jacobian of the universal family of genus-$g$ curves over $\mathcal{M}_g$.

math.AG↗
arXiv2023

Rational Points on Generic Marked Hypersurfaces

Over fields of characteristic zero, we show that for $n=1,d\geq4$ or $n=2,d\geq5$ or $n\geq3, d\geq 2n$, the generic $m$-marked degree-$d$ hypersurface in $\mathbb{P}^{n+1}$ admits the $m$ marked points as all the rational points. Over arbitrary fields, we show that for $n=1,d\geq4$ or $n\geq2, d\geq 2n+3$, the identiy map is the only rational self-map of the generic degree-$d$ hypersurface in $\mathbb{P}^{n+1}$.

math.AG↗
arXiv2023

Bounding the number of graph refinements for Brill-Noether existence

Let $G$ be a finite graph of genus $g$. Let $d$ and $r$ be non-negative integers such that the Brill-Noether number is non-negative. It is known that for some $k$ sufficiently large, the $k$-th homothetic refinement $G^{(k)}$ of $G$ admits a divisor of degree $d$ and rank at least $r$. We use results from algebraic geometry to give an upper bound for $k$ in terms of $g,d,$ and $r$.

math.AG↗
arXiv2019

Some properties of a Brauer class

Let $X$ be a smooth proper curve defined over a field $k$. The representability of the relative Picard functor is obstructed by a class $α\in\mathrm{Br}(\mathrm{Pic}_{X/k})$. We show the associated division algebra on $\mathrm{Pic}^0_{X/k}$ has natural involutions. We show the class $α$ splits at some height one points in $\mathrm{Pic}_{X/k}$.

math.AG↗
arXiv2019

Conics associated with totally degenerate curves

Let $k$ be a field. Let $X/k$ be a stable curve whose geometric irreducible components are smooth rational curves. Taking Stein factorization of its normalization, we get a conic. We show the conic is non-split in certain cases. As an application, we show for $g\geq3$, the period and index of the universal genus $g$ curve both equal to $2g-2$.

math.AG↗
arXiv2019

Closed points on cubic hypersurfaces

We generalize some results of Coray on closed points on cubic hypersurfaces. We show certain symmetric products of cubic hypersurfaces are stably birational.

math.AG↗