On Abel Maps of Stable Curves

摘要

We construct Abel maps for a stable curve X. Namely, for each one-parameter deformation of X to a smooth curve having regular total space and for each d ≥ 1, we construct by specialization a map α_x~d : X~d → P_x~d, where X is contained in X is the smooth locus and where P_x~d is the coarse moduli scheme for equivalence classes of degree-d "semibalanced" line bundles on semistable curves having X as a stable model. For d = 1, we show that α_x~1 extends to a map α_x~1 : X → P_x~1 and does not depend on the choice of the deformation. α_x~1 is injective.