Maximally symmetric stable curves II

Brill-Noether Theory of Maximally Symmetric Graphs

We analyze the Brill-Noether theory of trivalent graphs and multigraphs having largest possible automorphism group in a fixed genus. For trivalent multigraphs with loops of genus at least 3, we show that there exists a graph with maximal automorphism group which is Brill-Noether special. We prove similar results for multigraphs without loops of genus at least 6, as well as simple graphs of genus at least 7. This analysis yields counterexamples, in any sufficiently large genus, to a conjecture of Caporaso.

Maximally symmetric stable curves

We give a sharp bound on the number of automorphisms of a stable curve of a given genus and describe all curves attaining this bound.

极多节点六次曲面与线性行列式表示

Maximally nodal sextic surfaces and linear determinantal representations

研究极多节点六次曲面的节点结构，证明其存在35个对称半偶节点，从而得到线性行列式表示，且半二次sheaf为乌里奇sheaf。

The Fermat curve x^n+y^n+z^n: the most symmetric non-singular algebraic plane curve

A projective non-singular plane algebraic curve of degree d<=4 is called maximally symmetric if it attains the maximum order of the automorphism groups for complex non-singular plane algebraic curves of degree d. For d<=7, all such curves are known. Up to projectivities, they are the Fermat curve for d=5,7, see \cite{kmp1,kmp2}, the Klein quartic for d=4, see \cite{har}, and the Wiman sextic for d=6, see \cite{doi}. In this paper we work on projective plane curves defined over an algebraically closed field of characteristic zero, and we extend this result to every d>=8 showing that the Fermat …

G-stable rank of symmetric tensors and log canonical threshold

Shitov recently gave a counterexample to Comon's conjecture that the symmetric tensor rank and tensor rank of a symmetric tensor are the same. In this paper we show that an analog of Comon's conjecture for the G-stable rank introduced by Derksen is true: the symmetric G-stable rank and G-stable rank of a symmetric tensor are the same. We also show that the log-canonical threshold of a singularity is bounded by the G-stable rank of the defining ideal.

Stable degenerations of symmetric squares of curves

The main result of this article is that the component of the Alexeev-Kollár-Shepherd-Barron moduli space of stable surfaces parameterizing stable degenerations of symmetric squares of curves is isomorphic to the moduli space of stable curves. The moduli space of stable curves is used to ensure that any degeneration can be replaced by a degeneration to the symmetric square of a stable curve. These degenerations are not stable, however, but are partially resolved by the relative Hilbert scheme of length two subschemes in the fibers of a family of stable curves. The article gives a geometric desc…