arXiv subjects

Cindy Tan

Publications and source records attributed to Cindy Tan.

3 recordsLinked to original sources

arXiv2026

The complex projective plane as a ball quotient

In 1986, Deligne and Mostow constructed a ball quotient $\mathbb{B}^2 / \Gamma$ biholomorphic to the complex projective plane $\mathbb{P}^2$ whose branch locus is a line arrangement. In this paper, we show that if $\mathbb{P}^2$ is realized as a ball quotient whose branch divisor $D$ is an arrangement of smooth pairwise normal-crossing curves, then the orbifold $(\mathbb{P}^2,D)$ is isomorphic to either the Deligne-Mostow example or a certain degree 9 cover of it. This classification of "ball quotient structures" on $\mathbb{P}^2$ generalizes the $\mathbb{P}^1$ case due to Poincar\'e.

math.GT↗

arXiv2023

Smallest nonabelian quotients of surface braid groups

We give a sharp lower bound on the size of nonabelian quotients of the surface braid group $B_n(\Sigma_g)$ and classify all quotients that attain the lower bound: Depending on $n$ and $g$, a quotient of minimum order is either a symmetric group or a 2-step nilpotent $p$-group.

math.GT↗

arXiv2019

Spaces of generators for matrix algebras with involution

Let $k$ be an algebraically closed field of characteristic different from 2. Up to isomorphism, the algebra $\operatorname{Mat}_{n \times n}(k)$ can be endowed with a $k$-linear involution in one way if $n$ is odd and in two ways if $n$ is even. In this paper, we consider $r$-tuples $A_\bullet \in \operatorname{Mat}_{n\times n}(k)^r$ such that the entries of $A_\bullet$ fail to generate $\operatorname{Mat}_{n\times n}(k)$ as an algebra with involution. We show that the locus of such $r$-tuples forms a closed subvariety $Z(r;V)$ of $\operatorname{Mat}_{n\times n}(k)^r$ that is not irreducible. We describe the irreducible components and we calculate the dimension of the largest component of $Z(r;V)$ in all cases. This gives a numerical answer to the question of how generic it is for an $r$-tuple $(a_1, \dots, a_r)$ of elements in $\operatorname{Mat}_{n\times n}(k)$ to generate it as an algebra with involution.

math.RA↗