Principally polarized abelian variaties dimension two or three are Jacobian varieties

In this note we prove that a principally polarized abelian variety of dimension g ≤ 3 is the canonically polarized Jacobian variety of a (possibly reducible) algebraic curve; for g = 2 this result was proved by A. Weil. Using results of P. Deligne and D. Mumford on the irreducibility of moduli spaces of stable curves, we thus derive the irreducibility of certain moduli spaces of abelian varieties. For g ≥ 4 the number ½ g (g + 1) of moduli for abelian varieties of dimension g is bigger than the number 3g - 3 of moduli of algebraic curves of genus g; this explains the restriction on g we are making.