Related papers: Counting curves via degeneration
We investigate the universal Severi variety of rational curves on K3 surfaces, which parametrises irreducible rational curves in a fixed class on varying K3 surfaces of fixed genus. We investigate the conjecuted irreducibility of this space…
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 study a class of Calabi-Yau varieties that can be represented as a non-singular model of a double covering of $\mathbb P^3$ branched along certain octic surfaces. We compute Euler numbers of all constructed examples and describe their…
We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete…
We explain how to form a novel dataset of simply connected Calabi-Yau threefolds via the Gross-Siebert algorithm. We expect these to degenerate to Calabi-Yau toric hypersurfaces with certain Gorenstein (not necessarily isolated)…
We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) $X$ which is equipped with an involution $\imath$ preserving the holomorphic volume form. We define integers $n_{g,h}(\beta) $ which give a virtual count…
There are easy "polynomial" deformations of Calabi-Yau hypersurfaces in toric varieties performed by changing the coefficients of the defining polynomial of the hypersurface. In this paper, we explicitly constructed the ``non-polynomial''…
There is a large number of different ways of constructing Calabi-Yau manifolds, as well as related non-geometric formulations, relevant in string compactifications. Showcasing this diversity, we discuss explicit deformation families of…
We present two methods for studying fibrations of Calabi-Yau manifolds embedded in toric varieties described by single weight systems. We analyse 184,026 such spaces and identify among them 124,701 which are K3 fibrations. As some of the…
By considering mirror symmetry applied to conformal field theories corresponding to strings propagating in quintic hypersurfaces in projective 4-space, Candelas, de la Ossa, Green and Parkes calculated the ``number of rational curves on the…
We give a method for the computation of the plurigenera of a product-quotient manifold. We give two different types of applications to it: to the construction of Calabi-Yau threefolds and to the determination of the minimal model of a…
In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface $\fn$. We compute the number of such curves through the appropriate number of fixed general points on $\fn$, and the number of…
Recent results on duality between string theories and connectedness of their moduli spaces seem to go a long way toward establishing the uniqueness of an underlying theory. For the large class of Calabi-Yau 3-folds that can be embedded as…
Building on results of Clemens and Kley, we find criteria for a continuous family of curves in a nodal $K$-trivial threefold $Y_0$ to deform to a scheme of finitely many smooth isolated curves in a general deformation $Y_t$ of $Y_0$. As an…
We will use toric degenerations of the projective plane ${{\mathbb{P}}^ 2}$ to give a new proof of the triple points interpolation problems in the projective plane. We also give a complete list of toric surfaces that are useful as…
We present a novel way to classify Calabi-Yau threefolds by systematically studying their infinite volume limits. Each such limit is at infinite distance in Kahler moduli space and can be classified by an associated limiting mixed Hodge…
In this paper, we investigate the geometries associated with 3-forms of various orbital types on a symplectic 6-manifold. We demonstrate that certain unstable 3-forms, which naturally emerge from specific degenerations of Calabi-Yau…
Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of 2-dimensional torsion sheaves, enumerating pairs $Z\subset H$ in a Calabi-Yau threefold X. Here H is a member of a…
In order to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points, we construct a compactification of their moduli space. We term the latter a…
We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that…