Abel maps of Gorenstein curves

Abstract

For a Gorenstein curve X and a nonsingular point P∈ X, we construct Abel maps $A\:X\to J_X^1$ and $A_P\:X\to J_X^0$ , where J X i is the moduli scheme for simple, torsion-free, rank-1 sheaves on X of degree i. The image curves of A and A P are shown to have the same arithmetic genus of X. Also, A and A P are shown to be embeddings away from rational subcurves L⊂ X meeting $\overline{X-L}$ in separating nodes. Finally we establish a connection with Seshadri’s moduli scheme U X (1) for semistable, torsion-free, rank-1 sheaves on X, obtaining an embedding of A(X) into U X (1). Keywords Abel map, Torsion-free rank-1 sheaf, Compactified Jacobian, Gorenstein singularity Mathematics Subject Classification (2000) 14H40, 14H60