Welschinger invariants of real del Pezzo surfaces of degree ≥ 2

Abstract

International audienceWe compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree ≥ 2. We show that under some conditions, for such a surface X and a real nef and big divisor class D ∈ Pic(X), through any generic collection of −DK X − 1 real points lying on a connected component of the real part RX of X one can trace a real rational curve C ∈ |D|. This is derived from the positivity of appropriate Welschinger invariants. We furthermore show that these invariants are asymptotically equivalent, in the logarithmic scale, to the corresponding genus zero Gromov–Witten invariants. Our approach consists in a conversion of Shoval–Shustin recursive formulas counting complex curves on the plane blown up at seven points and of Vakil's extension of the Abramovich–Bertram formula for Gromov–Witten invariants into formulas computing real enumerative invariants

Relative enumerative invariants of real nodal del Pezzo surfaces

“…The development we suggest in this paper is two-fold: we extend the above cited result from [12] to counting curves through real collections of points containing any amount of Figure 1: Bifurcations of real rational curves pairs of complex conjugate points and, overall, to divisor classes that have nonzero intersection with E. In the case of real nodal del Pezzo surfaces of degree ≥ 2, the invariance of the count that we introduce takes place unconditionally (Theorem 1.4, Section 1.3). For real nodal del Pezzo surfaces of degree 1, the situation is unequal: on one hand, as we show in Section 6.2 for some divisor classes the invariance fails; on the other hand, as we show in Section 1.3, it takes place under certain restrictions on the divisor class (see Theorems 1.5 and 1.6).…”

“…For any dividing surgery f : X → ∆ a , the real part Pic R (X t ) of Pic(X t ), t ∈ (0, a), is identified with {D ∈ Pic R (Σ) : DE = 0}, see [12,Proposition 4.2]. In addition, (1) and such that DE = 0.…”

“…Note that, as it follows from the properties of Welschinger invariants of quadrics (see [12,Theorem 2.2]), one has RW 0 (P 2 , C 2 , F o , 0, 2kL) > 0 and RW 0 (P 2 , C 2 , F no , 0, dL) > 0 for any positive integers k and d; furthermore, log RW 0 (P 2 , C 2 , F o , 0, 2kL) = 4k log k + O(k), log RW 0 (P 2 , C 2 , F no , 0, dL) = 2d log d + O(d).…”

Welschinger invariants of blow-ups of symplectic 4-manifolds

“…Recall that there are 12 topological types of degree 2 real Del Pezzo surfaces. I. Itenberg, V. Kharlamov, and E. Shustin [24] computed the Welschinger invariants with purely real point constraints of all degree 2 real Del Pezzo surfaces. E. Brugallé [2], computed the Welschinger invariants with arbitrary point constraints of real degree 2 Del Pezzo surfaces with a non-orientable real part.…”

“…Using the degeneration technique, Itenberg-Kharlamov-Shustin [24] studied the positivity and asymptotics of Welschinger invariants of real del Pezzo surfaces of degree ≥ 2, and obtained some new real Caporaso-Harris type formulae and real analogues of Abramovich-Bertram-Vakil formula. In [6,7], Brugallé-Puignau applied the real version of symplectic sum formula to obtain a real version of Abramovich-Bertram-Vakil formula in the symplectic setting.…”