Compactified Picard stacks over the moduli space of curves with marked points

Abstract

For any d Z and g, n 0 such that 2g - 2 + n > 0, denote by Picd, g, n the stack whose sections over a scheme S consist of flat and proper families : C S of smooth curves of genus g, with n distinct sections si : S C and a line bundle L of relative degree d over C. Morphisms between two such objects are given by cartesian diagrams C

2 // C

S 1 // si II S s iU U such that si 1 = 2 si, 1 i n, together with an isomorphism 3 : L 2(L ). Picd, g, n is endowed with a natural forgetful map onto Mg, n and it is, of course, not complete. The present thesis consists of the construction of an algebraic stack Pd, g, n with a map d, g, n onto Mg, n with the following properties. (1) Pd, g, n and d, g, n fit in the following diagram; Picd, g, n

// Pd, g, n d, g, n

Mg, n

// Mg, n (2) d, g, n is universally closed; (3) Pd, g, n has a geometrically meaningful modular description. For n = 0 (and g 2), our compactification consists of a stack theoretical interpretation of Lucia Caporaso's compactification of the universal Picard variety over Mg. Then, for n > 0 and 2g-2+n > 1, we proceed by induction in the number of points following the guidelines of Knudsen's construction of Mg, n. 1