The Abel map for surface singularities I. Generalities and examples
Abstract
Abstract. Let (X, o) be a complex normal surface singularity. We fix one of its good resolutions X → X, an effective cycle Z supported on the reduced exceptional curve, and any possible (first Chern) class l′ ∈ H 2 (X , Z). With these data we define the variety ECal′ (Z ) of those effective Cartier divisors D supported on Z which determine a line bundles OZ (D) with first Chern class l′. Furthermore, we consider the affine space Picl′ (Z) ⊂ H1(OZ∗ ) of isomorphism classes of holomorphic line bundles with Chern class l′ and the Abel map cl′ (Z) : ECal′ (Z) → Picl′ (Z). The present manuscript develops the major properties of this map, and links them with the determination of the cohomology groups H1(Z,L), where we might vary the analytic structure (X, o) (supported on a fixed topological type/resolution graph) and we also vary the possible line bundles L ∈ Picl′ (Z). The case of generic line bundles of Picl′ (Z) and generic line bundles of the image of the Abel map will have priority roles. Rewriting the Abel map via Laufer duality based on integration of forms on divisors, we can make explicit the Abel map and its tangent map. The case of superisolated and weighted homogeneous singularities are exemplified with several details.
The theory has similar goals (but rather different techniques) as the theory of Abel map or Brill–Noether theory of reduced smooth projective curves.
Collections
Related items
Showing items related by title, author, creator and subject.

Three Observations on Commutators of Singular Integral Operators with BMO Functions
Pérez C.; RiveraRíos I.P. (AWMSpringer Series, Harmonic Analysis, Partial Differentail Equations, Complex Analysis, Banach Spaces, and Operator Theory, 20160701)Three observations on commutators of Singular Integral Operators with BMO functions are exposed, namely 1  The already known subgaussian local decay for the commutator, namely $\[\frac{1}{Q}\left\left\{x\in Q\, : ... 
On the controllability of Partial Differential Equations involving nonlocal terms and singular potentials
Biccari U. (20161212)In this thesis, we investigate controllability and observability properties of Partial Differential Equations describing various phenomena appearing in several fields of the applied sciences such as elasticity theory, ... 
Singularities of the Hilbert scheme of effective divisors
Dan A. (20170110)In this article, we study the Hilbert scheme of effective divisors in smooth hypersurfaces in $\mathbb{P}^3$, a topic not extensively studied. We prove that there exists such effective divisors $D$ satisfying the property: ...