Related papers: Genus one enumerative invariants in del-Pezzo surf…
Using the Semple bundle construction, we derive an intersection-theoretic formula for the number of simultaneous contacts of specified orders between members of a generic family of degree $d$ plane curves and finitely many fixed curves. The…
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.
This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus $g$ curves of fixed degree…
We determine the crossing number of polynomial size curve systems on standard surfaces, in terms of the genus, up to high precision.
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…
Orders on surfaces provide a rich source of examples of noncommutative surfaces. Other than some existence results, very little is known about the various moduli spaces that can be associated to them. Even fewer examples have been…
We consider the global symplectic classification problem of plane curves. First we give the exact classification result under symplectomorphisms, for the case of generic plane curves, namely immersions with transverse self-intersections.…
We compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree at least 2. We show that under some conditions, for such a surface $X$ and a real nef and big divisor class $D$,…
We give detailed descriptions of gluing pseudoholomorphic maps in symplectic geometry, especially in the presence of an obstruction bundle. The main motivation is to try to compare the symplectic and enumerative invariants of algebraic…
The invariance of the Welschinger numbers for real unnodal Del Pezzo surfaces, which we used for the enumeration of real rational curves on real toric Del Pezzo surfaces (see math.AG/0303378 and IMRN 49 (2003), 2639-2653), follows from…
We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of…
We enumerate the number of surfaces of degree $d$ in $P^3$ having a singular line of order $k$, passing through $\delta$ generic points (where $\delta$ is the dimension of moduli space of such surfaces).
For two oriented simple closed curves on a compact orientable surface with a connected boundary we introduce a simple computation of a value in the first homology group of the surface, which detects in some cases that the geometric…
We explain how to compute top-dimensional intersections of psi-classes on moduli spaces of m-stable curves. On the moduli spaces of Deligne-Mumford stable pointed curves of genus one, these intersection numbers are determined by two…
We consider systems of simple closed curves on surfaces and their total number of intersection points, their so-called crossing number. For a fixed number of curves, we aim to minimise the crossing number. We determine the minimal crossing…
The problem of map enumeration concerns counting connected spatial graphs, with a specified number $j$ of vertices, that can be embedded in a compact surface of genus $g$ in such a way that its complement yields a cellular decomposition of…
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study…
We define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on…
We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We calculate the Poincare polynomials of the…
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.…