The dimension of the image of the Abel map associated with normal surface singularities

2019/09/16 by Nagy, János, Némethi, András
#14J80 #32S05 #32S25 #32S50 #57M27 (Primary) 14Bxx #57R57 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1909.07023

Abstract

Let (X,o) be a complex normal surface singularity with rational homology sphere link and let \widetildeX be one of its good resolutions. Fix an effective cycle Z supported on the exceptional curve and also a possible Chern class l'∈ H²(\widetildeX,ℤ). Define \rm Eca^l'(Z) as the space of effective Cartier divisors on Z and c^l'(Z):\rm Eca^l'(Z)→ \rm Pic^l'(Z), the corresponding Abel map. In this note we provide two algorithms, which provide the dimension of the image of the Abel map. Usually, dim \rm Pic^l'(Z)=p_g, dim \rm Im (c^l'(Z)) and \rm codim \rm Im (c^l'(Z)) are not topological, they are in subtle relationship with cohomologies of certain line bundles. However, we provide combinatorial formulae for them whenever the analytic structure on \widetildeX is generic. The \rm codim \rm Im (c^l'(Z)) is related with \h¹(\widetildeX,L)\_L∈ \rm Im (c^l'(Z)); in order to treat the `twisted' family \h¹(\widetildeX,L₀⊗ L)\_L∈ \rm Im (c^l'(Z)) we need to elaborate a generalization of the Picard group and of the Abel map. The above algorithms are also generalized.

Related

• The Abel map for surface singularities I. Generalities and examples
• Invariants of relatively generic surface singularities II. Images of\n Abel maps
• The Abel map for surface singularities II. Generic analytic structure
• The Abel map for surface singularities III. Elliptic germs
• Class of images of Abel maps on normal surface singularities