Some open problems in computational algebraic geometry.

Abstract

The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry from a computational viewpoint. In this survey, we briefly describe some open problems of computational algebraic geometry which can be approached from such viewpoint. Some of the problems we discuss are the decomposition of Jacobians of genus two curves, automorphisms groups of algebraic curves and the corresponding loci in the moduli space of algebraic curves $\mathcal M_g$, inclusions among such loci, decomposition of Jacobians of algebraic curves with automorphisms, invariants of binary forms and the hyperelliptic moduli, theta functions of curves with automorphisms, etc. We decompose Jacobians of genus 3 curves with automorphisms and determine the inclusions among the loci for algebraic curves with automorphisms of genus 3 and 4.