Constructive algebraic geometry and random curves

Frank-Olaf Schreyer (Universitaet des Saarlandes)

If a moduli space (of curves) is unirational, then in principal we can compute a family which depends on freely on independent parameters. In practice the computation of such family is out of reach with the current computer algebra systems for most families. However, it is quite often feasible to compute the curve which correspond to a specific values of the free parameters in a given ground field. In particular for a finite ground field F_q, where we may choose the values randomly this leads to computation of random curves. The runtime of such algorithm lies in O(log q)^2), and work very well in practice. However even without the assumption of unirationality there can exist algorithms with produce random curves with runtime O(log q)^3). I will illustrate this approach for curves up to genus 15, with the emphasis on the geometric background of these approaches.