title: On the enumerative geometry of local curves
creator: Schimpf, Maximilian
subject: ddc-510
subject: 510 Mathematics
description: This thesis studies the Gromov-Witten and stable pair invariants of local  curves. In particular, we give a closed formula for the full descendent stable  pair theory of all (non-relative) local curves in terms of the Bethe roots of  the quantum intermediate long wave system. In the process, we derive a  new explicit description of these Bethe roots, which may be of independent  interest. We further deduce rationality, functional equation and a pole restriction for the descendent stable pair theory of local curves as conjectured  by Pandharipande and Pixton. Furthermore, we show how the Bethe roots  can be used to diagonalize the descedent invariants of the tube and give  explicit formulas for the first few descendents. On the Gromov-Witten side,  we conjecture that the Gromov-Witten theory of the local elliptic curve is  governed by quasi-Jacobi forms. Finally, we compute an infinite series of  special cases, which provides evidence for our conjecture.
date: 2025
type: Dissertation
type: info:eu-repo/semantics/doctoralThesis
type: NonPeerReviewed
format: application/pdf
identifier: https://archiv.ub.uni-heidelberg.de/volltextserver/37102/1/PhDthesis.pdf
identifier: DOI:10.11588/heidok.00037102
identifier: urn:nbn:de:bsz:16-heidok-371022
identifier:   Schimpf, Maximilian  (2025) On the enumerative geometry of local curves.  [Dissertation]     
relation: https://archiv.ub.uni-heidelberg.de/volltextserver/37102/
rights: info:eu-repo/semantics/openAccess
rights: http://archiv.ub.uni-heidelberg.de/volltextserver/help/license_urhg.html
language: eng