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Enumerating log rational curves on some toric varieties

Carl Lian, Naufil Sakran

TL;DR

The paper determines genus 0 fixed-domain logarithmic Tevelev degrees for toric varieties by an intersection-theoretic calculation on moduli spaces of naive log quasimaps, and proves a complete formula for the projective bundle X_{r,s,a}. It confirms Cela–Iribar López's tropical prediction in this family, with the logTev formula depending on tangency data via m_j, μ_{j,v}, and a power of a, under explicit inequalities; it also shows that the analogous conjecture for blow-ups of P^r at r points can fail, providing explicit non-enumerative behavior via excess intersection. The approach avoids tropical reduction, instead deriving explicit counts from a tower of projective bundles and transversality arguments, clarifying when fixed-domain log invariants are enumerative. Overall, the work situates logarithmic Tevelev degrees as robust invariants for projective bundles while delineating the limits of tropical predictions in related blow-up cases.

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 López. 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."

Enumerating log rational curves on some toric varieties

TL;DR

The paper determines genus 0 fixed-domain logarithmic Tevelev degrees for toric varieties by an intersection-theoretic calculation on moduli spaces of naive log quasimaps, and proves a complete formula for the projective bundle X_{r,s,a}. It confirms Cela–Iribar López's tropical prediction in this family, with the logTev formula depending on tangency data via m_j, μ_{j,v}, and a power of a, under explicit inequalities; it also shows that the analogous conjecture for blow-ups of P^r at r points can fail, providing explicit non-enumerative behavior via excess intersection. The approach avoids tropical reduction, instead deriving explicit counts from a tower of projective bundles and transversality arguments, clarifying when fixed-domain log invariants are enumerative. Overall, the work situates logarithmic Tevelev degrees as robust invariants for projective bundles while delineating the limits of tropical predictions in related blow-up cases.

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 López. 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."

Paper Structure

This paper contains 22 sections, 10 theorems, 111 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Counting curves and log curves
  3. Logarithmic Tevelev degrees
  4. New results
  5. Conventions
  6. Acknowledgments
  7. Moduli of log rational curves on ${\mathbb P}_{{\mathbb P}^r}(\mathcal{O}^s\oplus\mathcal{O}(-a))$
  8. The target variety
  9. Maps to $X_{r,s,a}$
  10. Naive log quasimaps to $X_{r,s,a}$
  11. Incidence loci
  12. Transversality
  13. Transversality for maps
  14. Twisting
  15. Vanishing sections
  16. ...and 7 more sections

Key Result

Theorem 1.3.1

Assume that we are in Situation sit:tev_X_rsa, so that the logarithmic Tevelev degrees of $X_{r,s,a}$ are defined. If all of the inequalities hold, then where When $a=0$, the expression $0^0$ is interpreted to equal 1. Otherwise, we have $\mathrm{logTev}^{X_{r,s,a}}_{\Gamma}=0$.

Figures (1)

  • Figure 1: Fan of $X=\mathrm{Bl}_{[0:1:0],[0:0:1]}({\mathbb P}^2)$

Theorems & Definitions (24)

  • Definition 1.2.1
  • Definition 1.2.2
  • Definition 1.2.3
  • Theorem 1.3.1
  • Definition 2.1.1
  • Definition 2.3.1
  • Definition 2.4.1
  • Definition 2.4.2
  • Proposition 3.0.1
  • Proposition 3.1.1
  • ...and 14 more