Pencils of quadrics and Gromov–Witten–Welschinger invariants of $$\mathbb {C}P^3$$ C P 3

Relative quantum cohomology

“…When n = 3, it is shown in [31] that the superpotential invariants recover Welschinger's invariants [32]. Thus, our calculations recover those of [1,2]. For arbitrary odd n, interior constraints restricted to odd powers of ω, and no boundary constraints, it is shown in [31] that the superpotential invariants recover the invariants of Georgieva [10].…”

A remark on Gromov-Witten-Welschinger invariants of $\mathbb{C} P^3\#\overline{\mathbb{C} P}^3$

“…This isomorphism gives a group structure on C. If C is considered as the quotient of C by a full rank lattice Λ for which p is identified with 0, the group structure induced by Φ is the same with the group structure induced from (C, +) by the quotient map. Φ also induces a series of isomorphisms Φ d between Pic d (C) and C (see [BG16b,…”

“…Kollár proposed pencils of quadrics of CP 3 can be used to compute the enumerative invariants of CP 3 [Kol15]. E. Brugallé and P. Georgieva completely computed the Welschinger invariants of CP 3 [BG16b]. They estiblished a relation between the GWW invariants of CP 3 and the GWW invariants of CP 1 × CP 1 .…”

“…and f is an immersion.Remark 4.3. By using[Wel05b, Lemma 4.3], one can prove, similar to the proof of[BG16b, Proposition 5.3], that the regular values ofev : M * 0,2d−k (CP 3 #CP 3 , d[l] − k[e]) → (CP 3 #CP 3 ) 2d−k contained in Ω 2d−k form a dense open subset Ω 2d−k ⊂ Ω 2d−k and the set C (x), x ∈ Ω 2d−k , contains exactly [pt], . .…”

Rational curves on Del Pezzo manifolds

“…The failure of property (3) for curves of positive genera is the main impediment to extending this method to arbitrary genus, but it is conceivable that excess intersection could make some headway. Finally, if possible, it would be very interesting to take the "direct product" of [4] and this paper, in order to compute the Gromov-Witten-Welschinger invariants of Fano threefolds of index 2.…”