Related papers: Quadratically Enriched Plane Curve Counting via Tr…
We prove that the number of rational points of bounded height on certain del Pezzo surfaces of degree 1 defined over Q grows linearly, as predicted by Manin's conjecture. Along the way, we investigate the average number of integral points…
A smooth tropical quartic curve has seven tropical bitangent classes. Their shapes can vary within the same combinatorial type of curve. We study deformations of these shapes and we show that the conditions determined by Cueto and Markwig…
We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on…
We prove a correspondence theorem for singular tropical surfaces in real three space, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we…
As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of…
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…
These notes are intended as an easy-to-read supplement to part of the background material presented in my talks on enumerative geometry. In particular, the numbers $n_3$ and $n_4$ of plane rational cubics through eight points and of plane…
Kontsevich's formula is a recursion that calculates the number of rational degree $d$ curves in $\mathbb{P}_{\mathbb{C}}^2$ passing through $3d-1$ general positioned points. Kontsevich proved it by considering curves that satisfy extra…
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…
We obtain a formula for the number of genus one curves with a fixed complex structure of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed…
We discuss, following Mikhalkin, Brugall\'e, and many others, the counting of curves on toric surfaces with prescribed genus, Newton polygon, and intersection pattern with the toric boundary divisor, both at assigned and unassigned points.…
We continue our quest for real enumerative invariants not sensitive to changing the real structure and extend the construction we uncovered previously for counting curves of anti-canonical degree $\leqslant 2$ on del Pezzo surfaces with…
We define a signed count of real rational pseudo-holomorphic curves appearing in a one-parameter family of real Spin symplectic K3 surfaces. We show that this count is an invariant of the deformation class of the family. In the case of a…
We study rational points on the elliptic surface given by the equation: $$y^2 = x^3 + AxQ(u,v)^2 + BQ(u,v)^3,$$ where $A,B\in \mathbb{Z}$ satisfy that $4A^3-27B^2\neq 0$ and $Q(u,v)$ is a positive-definite quadratic form. We prove…
We prove the "all-the-heights'' version of the Batyrev--Manin--Peyre conjecture for split quintic del Pezzo surfaces, both for counting rational points over global function fields in positive characteristic and for the motivic version over…
For any number field k, upper bounds are established for the number of k-rational points of bounded height on non-singular del Pezzo surfaces defined over k, which are equipped with suitable conic bundle structures over k.
We compute Gromov-Witten invariants of any genus for del Pezzo surfaces of degree $\ge2$. The genus zero invariants have been computed a long ago, Gromov-Witten invariants of any genus for del Pezzo surfaces of degree $\ge3$ have been found…
We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{P}^3$, whose image lies in a $\mathbb{P}^2$, passing through $r$ lines and $s$ points, where $r + 2s = 3d+2$. This can be viewed as a family version of…
In studying rational points on elliptic K3 surfaces of the form $f(t)y^2=g(x)$, where $f,g$ are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having…
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…