There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We consider M-theory compactification on Calabi-Yau threefolds. The recently discovered connection between the BPS states of wrapped M2 branes and the topological string amplitudes on the threefold is used both as a tool to compute topological string amplitudes at higher genera as well as to unravel the degeneracies and quantum numbers of BPS states. Moduli spaces of $k$-fold symmetric products of the wrapped M2 brane play a crucial role. We also show that the topological string partition function is the Calabi-Yau version of the elliptic genus of the symmetric product of $K3$'s and use the macroscopic entropy of spinning black holes in 5 dimensions to obtain new predictions for the asymptotic growth of the topological string amplitudes at high genera.

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/1999/0003/0005/ATMP-1999-0003-0005-a006.pdf

M theory, topological strings and spinning black holes

We describe local mirror symmetry from a mathematical point of view and make several A-model calculations using the mirror principle (localization). Our results agree with B-model computations from solutions of Picard-Fuchs differential equations constructed form the local geometry near a Fano surface within a Calabi-Yau manifold. We interpret the Gromov-Witten-type numbers from an enumerative point of view. We also describe the geometry of singular surfaces and show how the local invariants of singular surfaces agree with the smooth cases when they occur as complete intersections. † email: chi@math.harvard.edu, yau@math.harvard.edu ∗ email: klemm@ias.edu ∗∗ email: zaslow@math.nwu.edu

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/1999/0003/0003/ATMP-1999-0003-0003-a003.pdf

Local mirror symmetry: Calculations and interpretations

BPS states, string duality, and nodal curves on K3

We obtain a recursive formula answering the following question: How many irreducible, plane curves of degree d and (geometric) genus g pass through 3d-1+g general points in the plane? The formula is proved by studying suitable degenerations of plane curves.

Counting plane curves of any genus

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.

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

Nous montrons que les feuilletages holomorphes induits par les applications rationnelles quasi-homogenes remplissent les composantes irreductibles de l'espace F q (r, d) des feuilletages de codimension q et degre d de l'espace projectif ℙ r pour tout 1 ≤ q ≤ r — 2. Nous etudions la geometrie de telles composantes irreductibles. Nous montrons que ce sont des varietes rationnelles et calculons leur degre dans plusieurs cas.

Stability of foliations induced by rational maps

Given a family of divisors {/),} in a family of smooth varieties { Ys} and a sequence of integers mx, . . . , m,, we study the scheme parametrizing the points (s,yx, . . . ,y,) such that^, is a (possibly infinitely near) m,-fold point of D,. We obtain a general formula which yields, as special cases, the formula of de Jonquieres and other classical results of Enumerative Geometry. We also study the questions of finiteness and the multiplicities of the solutions.

https://www.ams.org/tran/1981-267-02/S0002-9947-1981-0626480-7/S0002-9947-1981-0626480-7.pdf

Counting divisors with prescribed singularities

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.

Enumeration of surfaces containing an elliptic quartic curve

Fil: Cukierman, Fernando Miguel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

/pdf/enumeration-of-surfaces-containing-an-elliptic-quartic-curve-5e1kc0dioy.pdf

Israel Vainsencher

Papers

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

Stability of foliations induced by rational maps

Counting divisors with prescribed singularities

Enumeration of surfaces containing an elliptic quartic curve

Enumeration of surfaces containing an elliptic quartic curve