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A Wall Crossing Formula for Motivic Enumerative Invariants

Andrés Jaramillo Puentes

TL;DR

The paper addresses refining enumerative counts of rational curves by working in the Grothendieck–Witt setting, yielding counts $N_{d,k}^{\mathbb{A}^1}(\mathcal{P})\in \mathrm{GW}(k)$ that encode both complex and real data via motivic multiplicities. It develops a wall-crossing formula over quadratic field extensions, using tropical geometry and trace maps in $\mathrm{GW}(k)$ to relate counts when point conditions switch between two $k$-rational points and a conjugate pair, with the rank vanishing and the signature recovering the real Welschinger relation in the appropriate specialization. The main contributions include (i) a precise wall-crossing identity connecting $\mathbb{A}^1$-counts across point-condition changes to blow-up invariants, (ii) a tropical–$\mathbb{A}^1$ framework enabling quadratically enriched correspondences, and (iii) explicit degree-$4$ motivic invariants of $\mathbb{P}^2_k$ over quadratic extensions expressed as polynomials in hyperbolic form $h$ and trace forms $\beta_i$. The results establish that these refined counts are monic polynomials in trace data of the prescribed degree and provide concrete invariants to compare across field extensions, with potential to extend to broader toric del Pezzo settings and correspondences.

Abstract

We prove an analog of the wall crossing formula for Welschinger invariants relating the difference of signed curve counting of real curves passing through configurations that differ by a pair of complex conjugated points, and a correspondence Welschinger invariant of the blow up. We prove this analogue for the motivic count of rational curves of fixed degree passing through a generic configuration of points, counted with a motivic multiplicity in the Grothendieck-Witt ring of a base field, extending the notions in the correspondence theorem between motivic invariants for $k$-rational point conditions and tropical curves. We use this formula to compute the degree 4 motivic enumerative invariants of the projective plane counting curves passing through configurations of points defined over quadratic extensions of a base field.

A Wall Crossing Formula for Motivic Enumerative Invariants

TL;DR

The paper addresses refining enumerative counts of rational curves by working in the Grothendieck–Witt setting, yielding counts that encode both complex and real data via motivic multiplicities. It develops a wall-crossing formula over quadratic field extensions, using tropical geometry and trace maps in to relate counts when point conditions switch between two -rational points and a conjugate pair, with the rank vanishing and the signature recovering the real Welschinger relation in the appropriate specialization. The main contributions include (i) a precise wall-crossing identity connecting -counts across point-condition changes to blow-up invariants, (ii) a tropical– framework enabling quadratically enriched correspondences, and (iii) explicit degree- motivic invariants of over quadratic extensions expressed as polynomials in hyperbolic form and trace forms . The results establish that these refined counts are monic polynomials in trace data of the prescribed degree and provide concrete invariants to compare across field extensions, with potential to extend to broader toric del Pezzo settings and correspondences.

Abstract

We prove an analog of the wall crossing formula for Welschinger invariants relating the difference of signed curve counting of real curves passing through configurations that differ by a pair of complex conjugated points, and a correspondence Welschinger invariant of the blow up. We prove this analogue for the motivic count of rational curves of fixed degree passing through a generic configuration of points, counted with a motivic multiplicity in the Grothendieck-Witt ring of a base field, extending the notions in the correspondence theorem between motivic invariants for -rational point conditions and tropical curves. We use this formula to compute the degree 4 motivic enumerative invariants of the projective plane counting curves passing through configurations of points defined over quadratic extensions of a base field.
Paper Structure (7 sections, 5 theorems, 37 equations, 6 figures)

This paper contains 7 sections, 5 theorems, 37 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Wall crossing formula for quadratic field extensions
  4. Tropical Geometry
  5. A1 Enumerative Geometry
  6. Proof of Theorem \ref{['thm:WCF']}
  7. Motivic plane invariants of degree 4 over quadratic field extensions

Key Result

Theorem 1.2

Let $k$ be a field of characteristic $0$ or characteristic greater than $d$. If $\Gamma\subset\mathbb{R}^2$ is a rational degree $d$ tropical curve passing through $\overline{\mathcal{P}}$, then under the canonical isomorphism $\operatorname{GW}(k\{\!\{t\}\!\})\cong \operatorname{GW}(k)$ the quadrat where the sum runs over all rational curves $C$ in $\mathbb{P}^2_{k\{\!\{t\}\!\}}$ passing through

Figures (6)

  • Figure 1: Lattice polygons corresponding to the blow up at a $k$-point fixed by the torus action.
  • Figure 2: Tropical curves of degree $\Delta_1$ passing through two $k$-points or a double point defined over a quadratic extension $k(\sqrt{c})$.
  • Figure 3: Tropical curves of degree $\Delta_2$ passing through two $k$-points or a double point defined over a quadratic extension $k(\sqrt{c})$, intersecting the bottom divisor in two $k$-points.
  • Figure 4: Tropical curves of degree $\Delta_2$ passing through two $k$-points or a double point defined over a quadratic extension $k(\sqrt{c})$, intersecting the bottom divisor in a double point.
  • Figure 5: Lattice polygons corresponding to quartic curves in the projective plane $\mathbb{P}^2$, and to curves in the first Hirzebruch surface $\Sigma_1$ of de-gree $4-2E$, respectively.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Definition 1.1
  • Theorem 1.2: JPP23
  • Corollary 1.3: JPP23
  • Theorem 1.4
  • Proposition 2.1: JPP22
  • Lemma 2.2
  • proof