Related papers: Tropical Methods for Counting Plane Curves -- Comp…
Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 5-folds. We find recursions for meeting numbers of genus 0 curves, and we determine the contributions of moving multiple covers of genus 0 curves to the…
We recursively compute the Gromov-Witten invariants of the Hilbert scheme of two points in the plane. By studying the space of stable maps and computing virtual contributions, we use these invariants to enumerate hyperelliptic plane curves…
The family of complex projective surfaces in projective three space of degree $d$ having precisely $\delta$ nodes as their only singularities has codimension $\delta$ in the linear system of surfaces of degree $d$ for sufficiently large $d$…
We first recall Solomon's relations for Welschinger's invariants counting real curves in real symplectic fourfolds, announced in \cite{Jake2} and established in \cite{RealWDVV}, and the WDVV-style relations for Welschinger's invariants…
We introduce tropical dual numbers as an extension of tropical semiring. By this innovation, one can work with honest ideals, instead of congruences, and recover the Euclidean topology on affine tropical spaces similar to Zariski's approach…
We give a constructive proof using tropical modifications of the existence of a family of real algebraic plane curves with asymptotically maximal numbers of even ovals.
For a convenient and Newton non-degenerate singularity, the Milnor number is computed from the complement of its Newton diagram in the first quadrant, so-called Kouchnirenko's formula. In this paper, we consider tropical curves dual to…
In the 1970s, William Tutte developed a clever algebraic approach, based on certain "invariants", to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks…
We study the geometry of varieties parametrizing degree d rational and elliptic curves in P^n intersecting fixed general linear spaces and tangent to a fixed hyperplane H with fixed multiplicities along fixed general linear subspaces of H.…
We present tools and definitions to study abstract tropical manifolds in dimension 2, which we call simply tropical surfaces. This includes explicit descriptions of intersection numbers of 1-cycles, normal bundles to some curves and…
We prove an analog of the wall crossing formula for Welschinger invariants relating the difference of signed curve counting of real curves passing through configurations that differ by a pair of complex conjugated points, and a…
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 study tropical line arrangements associated to a three-regular graph $G$ that we refer to as \emph{tropical graph curves}. Roughly speaking, the tropical graph curve associated to $G$, whose genus is $g$, is an arrangement of $2g-2$…
A $p$-adic version of Gromov-Witten invariants for counting plane curves of genus $g$ and degree $d$ through a given number of points is discussed. The multiloop version of $p$-adic string theory considered by Chekhov and others motivates…
The first part of this work constructs real positive-genus Gromov-Witten invariants of real-orientable symplectic manifolds of odd "complex" dimensions; the second part studies the orientations on the moduli spaces of real maps used in…
We introduce a new method to evaluate algebraic integrals over the simplex numerically. This new approach employs techniques from tropical geometry and exceeds the capabilities of existing numerical methods by an order of magnitude. The…
We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are…
We develop a version of Mikhalkin's lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a…
Given a curve defined over an algebraically closed field which is complete with respect to a nontrivial valuation, we study its tropical Jacobian. This is done by first tropicalizing the curve, and then computing the Jacobian of the…
We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are enriched with an information of multiplicity, sign, or argument. We obtain existence or uniqueness…