Families of Hyperelliptic Curves

Throughout this work we deal with a natural number g ≥ 2 and with an algebraically closed field k whose characteristic differs from 2. A hyperelliptic curve of genus g over k is a smooth curve of genus g, that is a double cover of the projective line P. The Riemann-Hurwitz formula implies that this covering should be ramified at 2g + 2 points. Because of this explicit description, hyperelliptic curves have been studied for a long time from different points of view. Among recent advances, we want to mention the determination of all the possible automorphism groups of hyperelliptic curves (see [BS86], [BGG 93], [Sha03]) as well as the extensive use of the Jacobian of hyperelliptic curves in cryptography (see [Sch85], [Can87], [Kob89], [Fre99], [Gau00], [Ked01], [Lan05], and the survey paper [JMS04]). In this paper we are interested in the moduli space Hg of hyperelliptic curves and in the moduli stack Hg of hyperelliptic curves, whose definitions we are going to briefly recall now. The MODULI SCHEME Hg of hyperelliptic curves is defined as