Enumerating log rational curves on some toric varieties

TL;DR

This paper computes genus 0 log Gromov-Witten invariants for certain toric varieties, confirming some conjectures via intersection theory on moduli spaces of log quasimaps.

Contribution

It provides a complete determination of these invariants for specific toric bundles and disproves a related conjecture for blow-ups, using direct intersection-theoretic methods.

Findings

Confirmed conjecture for projective bundles over projective space.

Disproved conjecture for blow-ups of projective space at points.

Established intersection-theoretic calculations on moduli spaces of naive log quasimaps.

Abstract

The genus 0, fixed-domain log Gromov-Witten invariants of a smooth, projective toric variety X enumerate maps from a general pointed rational curve to a smooth, projective toric variety passing through the maximal number of general points and with prescribed multiplicities along the toric boundary. We determine these invariants completely for the projective bundle X=P_{P^r}(O^s+O(-a)), proving a conjecture of Cela--Iribar L\'opez. A different conjecture when X is the blow-up of P^r at r points is disproven. Whereas the conjectures were predicted using tropical methods, we give direct intersection-theoretic calculations on moduli spaces of "naive log quasimaps."