Counting conics in complete intersections

Counting curves of any genus on P^2_7

We compute Gromov-Witten invariants of any genus for del Pezzo surfaces of degree $\ge2$. The genus zero invariants have been computed a long ago, Gromov-Witten invariants of any genus for del Pezzo surfaces of degree $\ge3$ have been found by Vakil. We solve the problem in two steps: (1) we consider the plane blown up at 6 points on a conic and one more point outside this conic (weak Fano surface), and, using techniques of tropical geometry, obtain a Caporaso-Harris type formula counting curves of any divisor class and genus subject to arbitrary tangency conditions with respect to the blown u…

On integral points of some Fano Threefolds and their Hilbert schemes of lines and conics

Let $X^o=\mathbb P^3\setminus D$ where $D$ is the union of two quadrics such that their intersection contains a smooth conic, or the union of a smooth quadric surface and two planes, or the union of a smooth cubic surface $V$ and a plane $Π$ such that the intersection $V\capΠ$ contains a line. In all these cases we show that the set of integral points of $X^o$ is potentially dense. We apply the above results to prove that integral points are potentially dense in some log-Fano or in some log-Calabi-Yau threefold.

Toric Intersection Theory for Affine Root Counting

Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate hyperplanes, these formulae are also the tightest possible upper bounds on the number of isolated roots. We also characterize, in terms of sparse resultants, precisely when these upper bounds are attained. Finally, we reformulate and extend some of the prior combinatorial results of the author on which subsets of coefficients must be chosen generically for o…

Counting ramified coverings and intersection theory on spaces of rational functions I (Cohomology of Hurwitz spaces)

The Hurwitz space is a compactification of the space of rational functions of a given degree. The Lyashko-Looijenga map assigns to a rational function the set of its critical values. It is known that the number of ramified coverings of CP^1 by CP^1 with prescribed ramification points and ramification types is related to the degree of the Lyashko--Looijenga map on various strata of the Hurwitz space. Here we explain how the degree of the Lyashko-Looijenga map is related to the intersection theory on this space. We describe the cohomology algebra of the Hurwitz space and prove several relations …

Rational Curves on the Space of Determinantal Nets of Conics

We describe the Hilbert scheme components parametrizing lines and conics on the space of determinantal nets of conics, N. As an application, we use the quantum Lefschetz hyperplane principle to compute the instanton numbers of rational curves on a complete intersection Calabi-Yau threefold in N. We also compute the number of lines and conics on some Calabi-Yau sections of non-decomposable vector bundles on N. The paper contains a brief summary of the A-model theory leading up to Givental-Kim's quantum Lefschetz hyperplane principle.

On Fano schemes of complete intersections

We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain pro-jective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete intersection is irregular of dimension at least 2, and for the Fano surfaces we deduce formulas for their holomorphic Euler characteristic.

完美域上与八条直线相交的圆锥曲线

Conics meeting eight lines over perfect fields

本文利用动机同伦理论中的欧拉类和局部度数，在任意完美域上给出了与8条一般位置直线相交的平面圆锥曲线数量的加权计数，并证明实圆锥曲线分为两个等规模的族。

打破超量交集中的对称性并计数多项式方程组的解

Breaking the symmetry in excess intersection and counting solutions of systems of polynomials

本文提出一种非对称相交重数方法，给出有序相交重数的显式描述，并借助牛顿图与 Bernstein-Kushnirenko 型估计逐步计数多项式方程组的孤立解，以 4 个一般球面的 12 条公切线为例验证了方法。

圆锥曲面映射与圆锥

Conics Quadrics Mapping & Cones

本文提出通过圆锥和二次曲面的射线获取五点圆锥和九点二次曲面的隐式方程的方法，并展示如何利用双四元数运动映射将共面五点置于圆锥上，同时探讨圆锥对称性对解的生成影响。

Counting twisted cubic curves on general complete intersections

This paper has been rendered obsolete by our newer eprint alg-geom/9411005 "Bott's formula and enumerative geometry", which is a considerably expanded version of the same paper, in spite of the change of titles. Please download alg-geom/9411005 instead, and update possible references.

Counting ramified coverings and intersection theory on Hurwitz spaces II (Local structure of Hurwitz spaces and combinatorial results)

The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these numbers belong to a very particular subalgebra of the algebra of power series.

Counting equilibria of the Kuramoto model using birationally invariant intersection index

Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appear naturally in many independent fields of science and engineering. A substantial amount of work has been devoted to understanding all possible synchronization configurations on a given network. In this setting, a key problem is to determine the total number of such configurations. Through an algebraic formulation, for tree and cycle graphs, we provide an upper bound on this number using the birationally invariant intersection index of a system of rational functions on a toric variety.

Rational points on pencils of conics and quadrics with many degenerate fibres

For any pencil of conics or higher-dimensional quadrics over the rationals, with all degenerate fibres defined over the rationals, we show that the Brauer-Manin obstruction controls weak approximation. The proof is based on the Hasse principle and weak approximation for some special intersections of quadrics, which is a consequence of recent advances in additive combinatorics.

Quantum cohomology of complete intersections

The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably simple form for complete intersections when the dimension is large enough w.r.t. the degree. As a reward we get a number of surprising enumerative formulas relating lines, conics and twisted cubics on X.