arXiv subjects

Israel Vainsencher

Publications and source records attributed to Israel Vainsencher.

6 recordsLinked to original sources

arXiv2023

Enumerative geometry of Legendrian foliations: a Tale of Contact

A contact distribution on projective three-space is defined by the 1-form $x_2dx_1-x_1dx_2+x_4dx_3-x_3dx_4$, up to a change of projective coordinates. The family of contact distributions is parameterized by the complement of the Pfaff-Plücker quadric in the projective 5-space of antisymmetric $4\times4$ matrices. A foliation of dimension 1 and degree $d$ is specified by a polynomial vector field $\sum{}p_i\partial_{x_i}, p_i$ homogeneous of degree $d$. The foliation is called Legendrian if tangent to some distribution of contact. Our goal is to give formulas for the dimensions and degrees of the varieties of Legendrian foliations, and of the varieties of foliations tangent to a pencil of planes.

math.AG↗

arXiv2018

Degree of the exceptional component of foliations in P3

The purpose of this work is to obtain the degree of the exceptional component of the space of holomorphic foliations of degree two and codimension one in P^3. We construct a parameter space as an explicit fiber bundle over the variety of complete flags. Using tools from equivariant intersection theory, especially Bott's formula, the degree is expressed as an integral over our parameter space.

math.AG↗

arXiv2014

Enumeration of surfaces containing an elliptic quartic curve

A very general surface of degree at least four in projective space of dimension three contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces of degree at least five which contain some elliptic quartic curve. We also compute the degree of the locus of quartic surfaces containing an elliptic quartic curve, a case not covered by that formula.

math.AG↗

arXiv2010

Polynomial vector fields with algebraic trajectories

It is known after Jouanolou that a general holomorphic foliation of degree $\geq2$ in projective space has no algebraic leaf. We give formulas for the degrees of the subvarieties of the parameter space of one-dimensional foliations that correspond to foliations endowed with some invariant subvariety of degree 1 or 2 and dimension $\geq1$.

math.AG↗

arXiv1996

Flatness of families induced by hypersurfaces on flag varieties

We answer a question posed by S. Kleiman concerning flatness of the family of complete quadrics. We also show that any flat family of hypersurfaces on Grassmann varieties induces a flat family of intersections with the corresponding flag variety.

alg-geom↗

arXiv1994

Enumeration of $n$-fold tangent hyperplanes to a surface

For each $1\leq n\leq6$ we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane sections of a generic $K3-$surface imbedded in \p{n} by a complete system of curves of genus $n$ as well as the number {\bf17,601,000} of rational ({\em singular}) plane quintic curves in a generic quintic threefold.

alg-geom↗