Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hyperelliptic classes are rigid and extremal in genus two

Vance Blankers

Abstract

We show that the class of the locus of hyperelliptic curves with $\ell$ marked Weierstrass points, $m$ marked conjugate pairs of points, and $n$ free marked points is rigid and extremal in the cone of effective codimension-($\ell + m$) classes on $\overline{\mathcal{M}}_{2,\ell+2m+n}$. This generalizes work of Chen and Tarasca and establishes an infinite family of rigid and extremal classes in arbitrarily-high codimension.

Hyperelliptic classes are rigid and extremal in genus two

Abstract

We show that the class of the locus of hyperelliptic curves with marked Weierstrass points, marked conjugate pairs of points, and free marked points is rigid and extremal in the cone of effective codimension-() classes on . This generalizes work of Chen and Tarasca and establishes an infinite family of rigid and extremal classes in arbitrarily-high codimension.

Paper Structure

This paper contains 3 sections, 6 theorems, 22 equations, 4 figures.

Table of Contents

  1. Preliminaries on $\overline{\mathcal{M}}_{g,n}$ and effective cycles
  2. Main theorem
  3. Higher genus

Key Result

Lemma 1.1

Let $g\geq 1$, $n\geq 2$, and $P\subset \{p_1,\dots,p_n\}$ such that $|P|\leq n-2$. Then for any $p_i,p_j\not\in P$ on $\overline{\mathcal{M}}_{g,n}$.

Figures (4)

  • Figure 1: On the left-hand side, the topological pictures of the general elements of $W_{2,P}$ (top) and $\gamma_{1,P}$ (bottom) in $\overline{\mathcal{M}}_{2,5}$ with $P = \{p_1,p_2,p_3\}$. On the right-hand side, the corresponding dual graphs.
  • Figure 2: An admissible cover in $\overline{Adm}_{2\xrightarrow{2} 0,t_1,\dots,t_6,u_{1\pm}}$ represented via dual graphs. In degree two the topological type of the cover is uniquely recoverable from the dual graph presentation.
  • Figure 3: The general element of $\overline{\mathcal{H}}_{2,2,0,3}$.
  • Figure 4: The point $[b_n]$ in $\overline{\mathcal{M}}_{0,\{t_1,\dots,t_5,u_{1},\dots,u_{n}\}}$.

Theorems & Definitions (14)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Definition 1.3
  • Lemma 1.4: chencoskun2014
  • Remark 1.5
  • Definition 2.1
  • Theorem 2.2
  • proof
  • ...and 4 more