Hypersurfaces In Projective Space Research Articles

On the Fukaya category of a Fano hypersurface in projective space

This paper is about the Fukaya category of a Fano hypersurface $X \subset \mathbb{CP}^n$. Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed-open string maps, weak proper Calabi-Yau structure, Abouzaid's split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface $X$: we construct a configuration of monotone Lagrangian spheres in $X$, and compute the associated disc potential. The result coincides with the Hori-Vafa superpotential for the mirror of $X$ (up to a constant shift in the Fano index $1$ case). As a consequence, we give a proof of Kontsevich's homological mirror symmetry conjecture for $X$. We also explain how to extract non-trivial information about Gromov-Witten invariants of $X$ from its Fukaya category.

Localized Standard Versus Reduced Formula and Genus 1 Local Gromov–Witten Invariants

For local Calabi–Yau (CY) manifolds which are total spaces of vector bundle over algebraic Goresky-Kottwitz-MacPherson (GKM) manifolds, we propose a formal definition of reduced genus 1 Gromov–Witten (GW) invariants, by assigning contributions from the refined decorated rooted trees. We show that this definition satisfies a localized version of the standard versus reduced (LSvR) formula, whose global version in the compact cases is due to Zinger. As an application, we prove the conjecture in a previous article on the genus 1 GW invariants of local CY manifolds which are total spaces of concave splitting vector bundles over projective spaces. In particular, we prove the mirror formulae for genus 1 GW invariants of |$K_{\mathbb {P}^{2}}$| and |$K_{\mathbb {P}^{3}}$|⁠, conjectured by Klemm, Zaslow, and Pandharipande. In the appendix, we derive the modularity of genus 1 GW invariants for the local |$\mathbb {P}^{2}$| as a consequence of the results on Ramanujan's cubic transformation. Inspired by the LSvR formula, we show that the ordinary genus 1 GW invariants of CY hypersurfaces in projective spaces can be computed by virtual localization and quantum hyperplane property after the contribution of a genus 1 vertex is replaced by a modified one.

Fano schemes of generic intersections and machine learning

We investigate Fano schemes of conditionally generic intersections, i.e. of hypersurfaces in projective space chosen generically up to additional conditions. Via a correspondence between generic properties of algebraic varieties and events in probability spaces that occur with probability one, we use the obtained results on Fano schemes to solve a problem in machine learning.

Orbifold quantum D-modules associated to weighted projective spaces

We construct in an abstract fashion the orbifold quantum cohomology (quantum orbifold cohomology) of weighted projective space, starting from the orbifold quantum differential operator. We obtain the product, grading, and intersection form by making use of the associated self-adjoint D-module and the Birkhoff factorization procedure. The method extends to the more difficult case of Fano hypersurfaces in weighted projective space. However, in contrast to the case of weighted projective space itself or a Fano hypersurface in projective space, a "small Birkhoff cell" can appear in the construction; we give an example of this phenomenon.

Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space

We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make explicit computations; and the introduction of certain graded categories of matrix factorizations mirror to the relative Fukaya category.

Multi-point virtual structure constants and mirror computation of $CP^2$-model

In this paper, we propose a geometrical approach to mirror computation of genus 0 Gromov-Witten invariants of CP^2. We use multi-point virtual structure constants, which are defined as intersection numbers of a compact moduli space of quasi maps from CP^1 to CP^2 with 2+n marked points. We conjecture that some generating functions of them produce mirror map and the others are translated into generating functions of Gromov-Witten invariants via the mirror map. We generalize this formalism to open string case. In this case, we have to introduce infinite number of deformation parameters to obtain results that agree with some known results of open Gromov-Witten invariants of CP^2. We also apply multi-point virtual structure constants to compute closed and open Gromov-Witten invariants of a non-nef hypersurface in projective space. This application simplifies the computational process of generalized mirror transformation.

Point Correspondences between <i>N</i> + 1 Hypersurfaces of Projective Spaces and (<i>N</i> + 1)-Webs

For a correspondence in question we establish a sequence of fundamental geometrical objects of the correspondence and find invariant normalizations of the first and second orders of all hupersurfaces under the correspondence. We single out main tensors of the correspondence and establish a connection between the geometry of point correspondences between n + 1 hypersurfaces of projective spaces and the theory of multidimensional (n + 1)-webs.

The number of reducible space curves over a finite field

"Most" hypersurfaces in projective space are irreducible, and rather precise estimates are known for the probability that a random hypersurface over a finite field is reducible. This paper considers the parametrization of space curves by the appropriate Chow variety, and provides bounds on the probability that a random curve over a finite field is reducible.

On projective-differential properties of point correspondences between three hypersurfaces

Point correspondences between three hypersurfaces of projective spaces are studied on the base of G. F. Laptev’s invariant methods [1] (Proc. of Moscow Math. Soc., 2, 275–382, 1953). We establish the basic equations and geometrical objects of the point correspondences in question. We construct invariant normalizations of the hypersurfaces, single out the basic tensors of the correspondences, and establish a connection between the geometry of correspondences and the theory of multidimensional 3-webs.

The Varieties of Tangent Lines to Hypersurfaces in Projective Spaces

For a hypersurface in a projective space, we consider the set of pairs of a point and a line in the projective space such that the line intersects the hypersurface at the point with a fixed multiplicity. We prove that this set of pairs forms a smooth variety for a general hypersurface.

Lines and osculating lines of hypersurfaces

We define systems of partial differential equations that govern the infinitesimal variation of lines, and osculating lines, through a point of a hypersurface in projective space. The work answers questions posed by J.-M. Hwang.

Smoothing nodal Calabi-Yau n -folds

Let X be an n-dimensional Calabi-Yau with ordinary double points, where n is odd. Friedman showed that for n=3 the existence of a smoothing of X implies a specific type of relation between homology classes on a resolution of X. (The converse is also true, due to work of Friedman, Kawamata and Tian.) We sketch a more topological proof of this result, and then extend it to higher dimensions. For n>3 the "Yukawa product" on the middle dimensional (co)homology plays an unexpected role. We also discuss a converse, proving it for nodal Calabi-Yau hypersurfaces in projective space.

Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian

We introduce various families of irreducible homaloidal hypersurfaces in projective space P r , for all r ⩾ 3 . Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of hypersurfaces that are dual to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hypersurfaces based on a certain degeneration of a generic Hankel matrix. The latter family fit non-classical versions of de Jonquières transformations. As a natural counterpoint, we broaden up aspects of the theory of Gordan–Noether hypersurfaces with vanishing Hessian determinant, bringing over some more precision into the present knowledge.

Numerical Calabi–Yau metrics

We develop numerical methods for approximating Ricci flat metrics on Calabi–Yau hypersurfaces in projective spaces. Our approach is based on finding balanced metrics and builds on recent theoretical work by Donaldson. We illustrate our methods in detail for a one parameter family of quintics. We also suggest several ways to extend our results.

Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity

The results we obtain in this article concern the hyperbolicity of very generic hypersurfaces in the 3-dimensional projective space: we show that the Kobayashi conjecture is true in this setting, as long as the degree of the hypersurface is greater than 18.

Twisted genus bounds for subvarieties of generic hypersurfaces

We prove some lower bounds on certain twists of the canonical bundle of a subvariety of a generic hypersurface in projective space. In particular we prove that the generic sextic threefold contains no rational or elliptic curves and no nondegenerate curves of genus 2.

Reducibility mod p of Hypersurfaces in Projective Spaces—An Application of Arithmetic Bézout

Given an absolutely irreducible horizontal hypersurface Z in a projective space over the ring of integers R of a number field, we give an explicit bound for the product of the norms of the prime ideals of R over which the fibre of Z becomes reducible. This bound is given as a function of a projective height of Z and is obtained using arithmetic intersection theory, in particular, an arithmetic Bézout theorem.

Chow groups of weighted hypersurfaces

We extend a result of to Esnault-Levine-Viehweg concerning the Chow groups of hypersurfaces in projective space to those in weighted projective spaces.

Nice Modular Varieties

We consider some more or less classical hypersurfaces in projective space, known to be birational to a quotient of the unit ball in the corresponding dimension by an arithmetic subgroup. We are interested in understanding the intersection of each such hypersurface with its Hessian from the point of view of arithmetic groups. In addition to unifying certain results found previously in the literature, we compute for four of these hypersurfaces the Hessian as well as its intersection with the hypersurface.

Subvarieties of general hypersurfaces in projective space

Subvarieties of general hypersurfaces in projective space