Appendix to "Welschinger invariant and enumeration of real rational curves"

Welschinger invariant and enumeration of real plane rational curves

Welschinger's invariant bounds from below the number of real rational curves through a given generic collection of real points in the real projective plane. We estimate this invariant using Mikhalkin's approach which deals with a corresponding count of tropical curves. In particular, our estimate implies that, for any positive integer $d$, there exists a real rational curve of degree $d$ through any collection of $3d-1$ real points in the projective plane, and, moreover, asymptotically in the logarithmic scale at least one third of the complex plane rational curves through a generic point coll…

Behavior of Welschinger Invariants under Morse Simplifications

We relate Welschinger invariants of a rational real symplectic 4-manifold before and after a Morse simplification (i.e deletion of a sphere or a handle of the real part of the surface). This relation is a consequence of a real version of Abramovich-Bertram formula which computes Gromov-Witten invariants by means of enumeration of $J$-holomorphic curves with a non-generic almost complex structure $J$. In addition, we give some qualitative consequences of our study, for example the vanishing of Welschinger invariants in some cases.

Recursive formulas for Welschinger invariants of the projective plane

Welschinger invariants of the real projective plane can be computed via the enumeration of enriched graphs, called marked floor diagrams. By a purely combinatorial study of these objects, we prove a Caporaso-Harris type formula which allows one to compute Welschinger invariants for configurations of points with any number of complex conjugated points.

Surgery of real symplectic fourfolds and Welschinger invariants

A surgery of a real symplectic manifold $X_{\mathbb R}$ along a real Lagrangian sphere $S$ is a modification of the symplectic and real structure on $X_{\mathbb R}$ in a neigborhood of $S$. Genus 0 Welschinger invariants of two real symplectic $4$-manifolds differing by such a surgery have been related in a previous work in collaboration with N. Puignau. In the present paper, we explore some particular situations where these general formulas greatly simplify. As an application, we complete the computation of genus 0 Welschinger invariants of all del~Pezzo surfaces, and of all $\mathbb R$-minim…

非结点实平面有理曲线的计数

Enumeration of non-nodal real plane rational curves

该研究探讨计数带非结点奇点的实有理平面曲线的不变量，证明唯一的此类不变量对应实有理三尖点四次曲线，且任意四对复共轭点的一般构型均存在一对这类曲线。

量子指标、阿诺德 - 罗赫林曲面与实枚举几何

Quantum index, Arnold-Rokhlin surfaces, and real enumerative geometry

研究关联用于定义复曲面实枚举不变量的两种韦尔申格型符号规则，其关系涉及量子指标与实曲线点集几何，还提出量子指标新定义用于非复曲面实代数曲线精细不变量枚举。