[Paper Review] Welschinger invariants of toric Del Pezzo surfaces with non-standard real structures

Eugeniĭ Shustin|ArXiv.org|May 27, 2006

Polynomial and algebraic computation21 references4 citations

TL;DR

This paper establishes a tropical formula for Welschinger invariants of four non-standard real toric Del Pezzo surfaces—${\mathbb{S}}^2$, ${\mathbb{S}}^2_{1,0}$, ${\mathbb{S}}^2_{2,0}$, and ${\mathbb{S}}^2_{0,2}$—using tropical enumerative geometry. It shows that the invariants are computed as weighted sums over combinatorial objects related to tropical curves, proving positivity for totally real configurations and confirming the existence of real rational curves in given linear systems.

ABSTRACT

The Welschinger invariants of real rational algebraic surfaces are natural analogues of the Gromov-Witten invariants, and they estimate from below the number of real rational curves passing through prescribed configurations of points. We establish a tropical formula for the Welschinger invariants of four toric Del Pezzo surfaces, equipped with a non-standard real structure. Such a formula for real toric Del Pezzo surfaces with a standard real structure (i.e., naturally compatible with the toric structure) was established by Mikhalkin and the author. As a consequence we prove that, for any real ample divisor $D$ on a surfaces $Σ$ under consideration, through any generic configuration of $c_1(Σ)D-1$ generic real points there passes a real rational curve belonging to the linear system $|D|$.

Motivation & Objective

• To compute Welschinger invariants for non-standard real toric Del Pezzo surfaces, which are not naturally compatible with the toric structure.
• To extend tropical enumerative geometry techniques beyond standard real toric Del Pezzo surfaces, where invariants were previously known.
• To establish a combinatorial formula expressing Welschinger invariants as weighted sums over tropical curves and deformation patterns.
• To prove the existence of real rational curves in given linear systems through generic configurations of real or conjugate imaginary points.
• To demonstrate that Welschinger invariants are positive for totally real configurations, implying the existence of real rational curves in the specified linear systems.

Proposed method

• Utilizes tropical enumerative geometry to express Welschinger invariants as sums over weighted combinatorial objects associated with tropical curves.
• Applies deformation patterns to limit curves, distinguishing between real and conjugate pairs based on component weights and nodal parity.
• Imposes point conditions via valuation and initial conditions on parameters, ensuring transversality and existence of real rational curves.
• Uses the fact that even-weighted components corresponding to real points contribute in pairs with opposite signs, thus canceling out in the invariant.
• For odd-weighted components at real points, ensures consistent sign (parity of solitary nodes) across all deformation pattern choices.
• Relies on results from [17] and [18] to ensure smooth dependence of curves on parameters and transversality of point conditions.

Experimental results

Research questions

• RQ1 Can Welschinger invariants be computed for non-standard real toric Del Pezzo surfaces using tropical geometry?
• RQ2 What is the combinatorial structure of tropical curves that correspond to real rational curves on these surfaces?
• RQ3 How do deformation patterns and nodal parity affect the sign of the Welschinger invariant in non-standard real structures?
• RQ4 Under what conditions does the Welschinger invariant remain positive, and what does this imply for the existence of real rational curves?
• RQ5 Is there a uniform asymptotic behavior of Welschinger invariants comparable to that of Gromov-Witten invariants on these surfaces?

Key findings

• The Welschinger invariant for any real ample line bundle $\mathcal{L}$ on ${\mathbb{S}}^2$, ${\mathbb{S}}^2_{1,0}$, ${\mathbb{S}}^2_{2,0}$, or ${\mathbb{S}}^2_{0,2}$ is given by a tropical formula involving weighted sums over combinatorial objects.
• For totally real configurations of $c_1(\Sigma)D - 1$ points, the Welschinger invariant is positive, implying the existence of at least one real rational curve in $|D|$.
• When all components corresponding to real points have odd weight, the number of contributing real rational curves is $\prod_{j=1}^{r''_1} (w(G'_j))^2$, all with the same sign determined by nodal parity.
• Even-weighted components at real points lead to cancellation in the invariant due to paired deformation patterns with opposite nodal parity.
• The asymptotic growth rate of the logarithm of the Welschinger invariant matches that of the Gromov-Witten invariant, satisfying $\lim_{n\to\infty} \frac{\log W_0(\Sigma, \mathcal{L}^{\otimes n})}{n\log n} = -c_1(\mathcal{L})K_\Sigma$.
• The invariant is independent of the choice of generic configuration due to Welschinger’s invariance theorem, confirmed via tropical methods.