Blowing up quantum weighted projective planes

Hilbert Schemes for Quantum Planes are Projective

We show that Hilbert schemes for the quantum plane are projective. We also show that some collections of torsion sheaves are bounded. Throughout this paper, all objects will be defined over a fixed ground field k.

Classification of Weighted Graphs up to Blowing-up and Blowing-down

We classify weighted forests up to the blowing-up and blowing-down operations which are relevant for the study of algebraic surfaces. The word “graph” in this text means a finite undirected graph such that no edge connects a vertex to itself and at most one edge joins any given pair of vertices. A weighted graph is a graph in which each vertex is assigned an integer (called its weight). Two ope...

Mirror Symmetry for Weighted Projective Planes and Their Noncommutative Deformations

Orbifold Quantum Cohomology of Weighted Projective Spaces

In this article, we prove the following results. • We show a mirror theorem : the Frobenius manifold associated to the orbifold quantum cohomology of weighted projective space is isomorphic to the one attached to a specific Laurent polynomial, • We show a reconstruction theorem, that is, we can reconstruct in an algorithmic way the full genus 0 Gromov-Witten potential from the 3-point invariants.

Smooth Projective Planes

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to CP. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plane curves are smooth collineations, and prove a variety of results analogous to the theory ...