Author:
Nagy János,Némethi András
Abstract
AbstractLet (X, o) be a complex normal surface singularity with rational homology sphere link and let $$\widetilde{X}$$
X
~
be one of its good resolutions. Consider an effective cycle Z supported on the exceptional curve and the isomorphism classes $$\mathrm{Pic}(Z)$$
Pic
(
Z
)
of line bundles on Z. The set of possible values $$h^1(Z,\mathcal {L})$$
h
1
(
Z
,
L
)
for $$\mathcal {L}\in \mathrm{Pic}(Z)$$
L
∈
Pic
(
Z
)
can be understood in terms of the dimensions of the images of the Abel maps, as subspaces of $$\mathrm{Pic}(Z)$$
Pic
(
Z
)
. In this note we present two algorithms, which provide these dimensions. Usually, the dimension of $$\mathrm{Pic}(Z)$$
Pic
(
Z
)
and of the dimension of the image of the Abel maps are not topological. However, we provide combinatorial formulae for them in terms of the resolution graph whenever the analytic structure on $$\widetilde{X}$$
X
~
is generic.
Funder
ELKH Alfréd Rényi Institute of Mathematics
Publisher
Springer Science and Business Media LLC
Subject
General Physics and Astronomy,General Mathematics
Reference26 articles.
1. Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves, vol. 1. Grundlehren der Mathematischen Wissenschaften 267. Springer, New York (1985)
2. Eisenbud, D., Green, M., Harris, J.: Cayley–Bacharach theorems and conjectures. Bull. AMS 33(3), 295–324 (1996)
3. Flamini, F.: Lectures on Brill–Noether theory, in Muk, J.-M., Kim, Y.R. (eds) Proceedings of the Workshop "Curves and Jacobians". Korea Institute for Advanced Study, pp. 1–20 (2011)
4. Grothendieck, A.: Fondements de la géométrie algébrique, [Extraits du Séminaire Bourbaki 1957–1962]. Secrétariat mathématique, Paris (1962)
5. Jensen, D., Ranganathan, D.: Brill–Noether theory for curves of fixed gonality. Forum Math. Pi 9(e1), 1–33 (2021)