Counting rational curves of arbitrary shape in projective spaces

Counting Rational Curves of Arbitrary Shape in Projective Spaces

Enumeration of One-Nodal Rational Curves in Projective Spaces

We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. The formula involves intersections of tautological classes on moduli spaces of stable rational maps. We combine the methods and results from three different papers.

Using Symmetry to Count Rational Curves

The recent “close encounter” between enumerative algebraic geometry and theoretical physics has resulted in many new applications and new techniques for counting algebraic curves on a complex projective manifold. For example, string theorists demonstrated that generating functions built from counting curves can have completely unexpected relationships with other geometric constructions, via mir...

Projective and affine symmetries and equivalences of rational curves in arbitrary dimension

We present a new algorithm to decide whether two rational parametric curves are related by a projective transformation and detect all such projective equivalences. Given two rational curves, we derive a system of polynomial equations whose solutions define linear rational transformations of the parameter domain, such that each transformation corresponds to a projective equivalence between the t...

Gromov–witten Classes, Quantum Cohomology, and Enumerative Geometry

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their cor...