NORMAL FORMS FOR PERIOD MATRICES OF REAL CURVES OF GENUS-2 AND GENUS-3

In general there is no normalized form for the period matrix of an alg ebraic curve. For real curves of genus 2 and 3, however, such a form d oes exist (up to a finite, but explicit, ambiguity for genus 3). We de scribe this form and give an algorithm for computing it. This algorith m is in fact an algorithm for computing the Minkowski reduced form of a quadratic form of dimension 2 or 3. The ideas underlying the algorit hm are based on the geometry of the moduli space of such curves.