An imaginary refined count for some real rational curves

Imaginary automorphisms on real hyperelliptic curves

A real hyperelliptic curve X is said to be Gaussian if there is an automorphism α : XC → XC such that α = [−1]C ◦ α, where [−1] denotes the hyperelliptic involution on X. Gaussian curves arise naturally in several contexts, for example when one studies real Jacobians. In the present paper we study the properties of Gaussian curves and we describe their moduli spaces. MSC 2000: 14H15, 14H37, 14P...

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...

Real Rational Curves in Grassmanniansfrank

Algorithms for Rational Real Algebraic Curves

In this paper, we study fundamental properties of real curves, especially of rational real curves, and we derive several algorithms to decide the reality and rationality of curves in the complex plane. Furthermore, if the curve is real and rational, we determine a real parametrization. More precisely, we present a reality test algorithm for plane curves, and three different types of real parame...

Maximally Inflected Real Rational Curves

We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert calculus, and their study has consequences for the important Shapiro and Shapiro conjecture in the real Schubert calculus. We establish restrictions on the number of real nodes of such curves and construc...