# A characterization of the U(Omega,m) sets of a hyperelliptic curve as Omega and m vary

## TL;DR

This paper characterizes the possible sets U(Ω,m) associated with hyperelliptic curves as the period matrix Ω and marking m vary, showing they encompass all configurations allowed by prior theoretical constraints.

## Contribution

It provides a comprehensive description of how the sets U(Ω,m) vary and range over all permissible configurations for hyperelliptic curves with different period matrices and markings.

## Key findings

- U(Ω,m) sets vary over all configurations allowed by Poor's argument.
- The variation of U(Ω,m) covers the entire prescribed set of possibilities.
- The results connect the geometry of hyperelliptic curves with the combinatorial structure of U(Ω,m).

## Abstract

In this article we consider a certain distinguished set $U(\Omega,m) \subseteq \{1,2,\ldots,2g+1,\infty\}$ that can be attached to a marked hyperelliptic curve of genus $g$ equipped with a small period matrix $\Omega$ for its polarized Jacobian. We show that as $\Omega$ and the marking $m$ vary, this set ranges over all possibilities prescribed by an argument of Poor.