The Abel map for surface singularities III. Elliptic germs

2019/02/20 by Nagy, János, Némethi, András
#14J80 (Secondary) #32S05 #32S25 #32S50 (Primary) 14Bxx #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.07493

Abstract

If (\widetildeX,E)→ (X,o) is the resolution of a complex normal surface singularity and c₁:\rm Pic(\widetildeX)→ H²(\widetildeX,\mathbb Z) is the Chern class map, then \rm Pic^l'(\widetildeX):= c₁^-1(l') has a (Brill--Noether type) stratification W_{l', k}:= \\mathcal L∈ \rm Pic^l'(\widetildeX) : h¹(\mathcal L)=k\. In this note we determine it for elliptic singularities together with the stratification according to the cycle of fixed components. For elliptic singularities we also characterize the End Curve Condition and Weak End Curve Condition in terms of the Abel map, we provide several characterization of them, and finally we show that they are equivalent.

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