Enumeration of Genus-Two Curves with a Fixed Complex Structure in P^2 and P^3

Enumeration of Genus-Three Plane Curves with a Fixed Complex Structure

We give a practical formula for counting irreducible nodal genus-three plane curves that a fixed generic complex structure on the normalization. As an intermediate step, we enumerate rational plane curves that have a $(3,4)$-cusp.

Enumeration of One-Nodal Rational Curves in Projective Spaces

We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. We combine the methods and results from three different papers.

Enumeration of real curves in CP^{2n-1} and a WDVV relation for real Gromov-Witten invariants

We establish a homology relation for the Deligne-Mumford moduli spaces of real curves which lifts to a WDVV-type relation for real Gromov-Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these invariants. For many real symplectic manifolds, these results reduce all genus 0 real invariants with conjugate pairs of constraints to genus 0 invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all c…

Intersections of Tautological Classes on Blowups of Moduli Spaces of Genus-One Curves

We give two recursions for computing top intersections of tautological classes on blowups of moduli spaces of genus-one curves. One of these recursions is analogous to the well-known string equation. As shown in previous papers, these numbers are useful for computing genus-one enumerative invariants of projective spaces and Gromov-Witten invariants of complete intersections.

射影平面爆破中的曲线热带计数

Tropical enumeration of curves in blowups of the projective plane

提出一种利用tropical曲线和相对Gromov-Witten不变量递归计算射影平面所有爆破的Gromov-Witten不变量的方法，该递归公式不同于已有在零亏格和无爆破情形下的公式。

Lower Bounds for Enumerative Counts of Positive-Genus Real Curves

We transform the positive-genus real Gromov-Witten invariants of many real-orientable symplectic threefolds into signed counts of curves. These integer invariants provide lower bounds for counts of real curves of a given genus that pass through conjugate pairs of constraints. We conclude with some implications and related conjectures for one- and two-partition Hodge integrals.