Residuation of Linear Series and The Effective Cone of C_d

TL;DR

This paper investigates the geometry of divisors on symmetric powers of general and hyperelliptic curves, providing new descriptions of effective cones, volume bounds, and base locus properties for various genera and degrees.

Contribution

It offers a complete description of the effective cone of $C_{g-1}$ for general curves and hyperelliptic curves, along with bounds on cones and volume computations for certain degrees.

Findings

Complete description of the effective cone of $C_{g-1}$ for general curves.

Bounds for effective and movable cones of $C_{d}$ in specified ranges.

Construction of divisors with non-equidimensional stable base locus.

Abstract

We obtain new information about divisors on the $d-$th symmetric power $C_{d}$ of a general curve $C$ of genus $g \geq 4.$ This includes a complete description of the effective cone of $C_{g-1}$ and a partial computation of the volume function on one of its non-nef subcones, as well as new bounds for the effective and movable cones of $C_{d}$ in the range $\frac{g+1}{2} \leq d \leq g-2.$ We also obtain, for each $g \geq 5,$ a divisor on $C_{g-1}$ with non-equidimensional stable base locus. For a general hyperelliptic curve $C$ of genus $g,$ we obtain a complete description of the effective cone of $C_{d}$ for $2 \leq d \leq g$ and an integral divisor on $C_{g-1}$ which has non-integral volume whenever $g$ is not a power of 2.