Related papers: Refined enumerative invariants and mixed Welsching…
In arXiv:1505.04338(4), G. Mikhalkin introduced a refined count for the real rational curves in a toric surface which pass through certain conjugation invariant set of points on the toric boundary of the surface. Such a set consists of real…
We introduce enumerative invariants of real del Pezzo surfaces that count real rational curves belonging to a given divisor class, passing through a generic conjugation-invariant configuration of points and satisfying preassigned tangency…
In 2015, G.~Mikhalkin introduced a refined count for real rational curves in toric surfaces. The counted curves have to pass through some real and complex points located on the toric boundary of the surface, and the count is refined…
We address the problem of existence of refined (i.e., depending on a formal parameter) tropical enumerative invariants, and we present two new examples of a refined count of rational marked tropical curves. One of the new invariants counts…
We prove that the G\"ottsche-Schroeter and Schroeter-Shustin refined invariants specialize at $q=1$ to the enumeration of rational, resp. elliptic complex curves on arbitrary toric surfaces matching constraints that consist of points and of…
Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal…
We introduce a \textit{quantum index} for oriented real curves inside toric varieties. This quantum index is related to the computation of the area of the amoeba of the curve for some chosen 2-form. We then make a refined signed count of…
We define refined invariants which "count" nodal curves in sufficiently ample linear systems on surfaces, conjecture that their generating function is multiplicative, and conjecture explicit formulas in the case of K3 and abelian surfaces.…
The Welschinger invariants of real rational algebraic surfaces count real rational curves which represent a given divisor class and pass through a generic conjugation-invariant configuration of points. No invariants counting real curves of…
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…
We prove that the quadratically enriched count of rational curves in a smooth toric del Pezzo surface passing through $k$-rational points and pairs of conjugate points in quadratic field extensions $k\subset k(\sqrt{d_i})$ can be determined…
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…
We give a tropical description of the counting of real log curves in toric degenerations of toric varieties. We treat the case of genus zero curves and all non-superabundant higher-genus situations. The proof relies on log deformation…
We introduce new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from the appropriate enumeration of real elliptic curves. These invariants admit a refinement (according to the quantum index)…
Tropical refined invariants of toric surfaces constitute a fascinating interpolation between real and complex enumerative geometries via tropical geometry. They were originally introduced by Block and G\"ottsche, and further extended by…
We introduce new invariants of a class of toric surfaces (including the projective plane) that arise from appropriate enumeration of real curves of genus one and two. These invariants admit a refinement similar to the one introduced by…
We suggest an invariant way to enumerate nodal and nodal-cuspidal real deformations of real plane curve singularities. The key idea is to assign Welschinger signs to the counted deformations. Our invariants can be viewed as a local version…
We propose a geometric interpretation of Block and G\"ottsche's refined tropical curve counting invariants in terms of virtual $\chi_{-y}$-specializations of motivic measures of semialgebraic sets in relative Hilbert schemes. We prove that…
We prove a $q$-refined tropical correspondence theorem for higher genus descendant logarithmic Gromov--Witten invariants with a $\lambda_g$ class in toric surfaces. Specifically, a generating series of such logarithmic Gromov--Witten…
Since the first famous correspondence theorem by Mikhalkin appeared in 2005, tropical geometry has allowed a parallel treatment of real and complex counting problems. A prime example are the genus 0 Gromov-Witten invariants of the plane…