Floor diagrams and enumerative invariants of line bundles over an elliptic curve

We use the tropical geometry approach to compute absolute and relative enumerative invariants of complex surfaces which are $\mathbb {C} P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block–Göttsche refined multipl...

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Bibliographic Details
Published in:Compositio mathematica 2023-08, Vol.159 (8), p.1741-1790
Main Author: Blomme, Thomas
Format: Article
Language:English
Subjects:
Algorithms
Curves
Geometry
Invariants
Modularity
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Summary:We use the tropical geometry approach to compute absolute and relative enumerative invariants of complex surfaces which are $\mathbb {C} P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block–Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invariants, and the quasi-modularity of their generating series.
ISSN:0010-437X
1570-5846
DOI:10.1112/S0010437X23007285