The Abel map for surface singularities III. Elliptic germs

TL;DR

This paper studies the stratification of the Picard group for elliptic surface singularities, characterizes conditions related to the Abel map, and proves the equivalence of the End Curve and Weak End Curve Conditions.

Contribution

It determines the stratification of the Picard group for elliptic singularities and characterizes the End Curve Conditions via the Abel map, establishing their equivalence.

Findings

Stratification of Picard group for elliptic singularities determined.

Characterizations of End Curve and Weak End Curve Conditions provided.

Proved the equivalence of End Curve and Weak End Curve Conditions.

Abstract

If $(\widetilde{X},E)\to (X,o)$ is the resolution of a complex normal surface singularity and $c_1:{\rm Pic}(\widetilde{X})\to H^2(\widetilde{X},{\mathbb Z})$ is the Chern class map, then ${\rm Pic}^{l'}(\widetilde{X}):= c_1^{-1}(l')$ has a (Brill--Noether type) stratification $W_{l', k}:= \{{\mathcal L}\in {\rm Pic}^{l'}(\widetilde{X})\,:\, h^1({\mathcal L})=k\}$. 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.

Equations

$$0\to {\mathrm{Pic}}^0(\tilde{X})\simeq H^1({\mathcal{O}}_{\tilde{X}})\to \mathrm{Pic}(\tilde{X})\stackrel{c_1}{⟶}{L}^{\prime }:=H^2(\tilde{X},\mathbb{Z})\to 0.$$

Full text

1. Introduction

1.1.

Recall that the classical Brill–Noether problem for curve is the following. Let $C$ be a smooth projective (complex) curve and let $c_{1}:{\rm Pic}(C)\to H^{2}(C,\mathbb{Z})=\mathbb{Z}$ be the first Chern class map. Set ${\rm Pic}^{l}(C):=c_{1}^{-1}(l)$. Then one considers the stratification of ${\rm Pic}^{l}(C)$ according to the $h^{1}$–value, namely, $W_{l,k}:=\{\mathcal{L}\in{\rm Pic}^{l}(C)\,:\,h^{1}(\mathcal{L})=k\}$. The problem is to determine the values $(l,k)$ for which $W_{l,k}$ is non–empty and in such cases to describe the topology of the different strata $W_{l,k}$. (This depends heavily on the analytic structure of $C$.) For details see e.g. [ACGH, Flamini].

1.1.1**.**

For complex normal surface singularities the analogue question can be formulated as follows. Let $(X,o)$ be such a singularity and let us fix a resolution $\phi:(\widetilde{X},E)\to(X,o)$. We will assume that the link is a rational homology sphere, equivalently, that the dual resolution graph $\Gamma$ is a tree of ${\mathbb{P}}^{1}$’s. Then one has the exponential exact sequence

[TABLE]

Here $L^{\prime}$ might serve also as the dual lattice of $L:=H_{2}(\widetilde{X},\mathbb{Z})$ freely generated by the irreducible exceptional divisors and endowed with the negative definite intersection form. Then for any possible Chern class $l^{\prime}\in L^{\prime}$ set ${\rm Pic}^{l^{\prime}}(\widetilde{X}):=c_{1}^{-1}(l^{\prime})$. (Note that while ${\rm Pic}^{l}(C)$ for a smooth curve is a compact complex torus, in the surface singularity case ${\rm Pic}^{l^{\prime}}(\widetilde{X})$ is a (non–compact) affine space $\mathbb{C}^{p_{g}}$, where $p_{g}$ is the geometric genus of $(X,o)$.) Following [NNI, NNII] we consider the stratification $W_{l^{\prime},k}:=\{\mathcal{L}\in{\rm Pic}^{l^{\prime}}(\widetilde{X})\,:\,h^{1}(\mathcal{L})=k\}$. Again, the goal is to describe the spaces $W_{l^{\prime},k}$. In general, they depend in a rather arithmetical way on the combinatorics of the resolution graph $\Gamma$ and also on the analytic structure supported on $\Gamma$. Usually the spaces $W_{l^{\prime},k}$ are not open, nor closed, not even quasi–projective. They might be nonlinear, their closure might be even singular. Though in theory of singularities several results are known for the possible $h^{1}(\mathcal{L})$–values (vanishing theorems, coarse topological bounds), a systematic study of the spaces $W_{l^{\prime},k}$ was missing, in the series of article (starting with [NNI, NNII] and the present one) the authors aim to fill in this necessity.