Curves generating extremal rays in blowups of weighted projective planes

An Extremal Characterization of Projective Planes

In this article, we prove that amongst all n by n bipartite graphs of girth at least six, where n = q + q + 1 ≥ 157, the incidence graph of a projective plane of order q, when it exists, has the maximum number of cycles of length eight. This characterizes projective planes as the partial planes with the maximum number of quadrilaterals.

On cubic curves in projective planes of characteristic two

The aim of this paper is to examine various interesting results from the theory of general cub~c curves in projective planes of characteristic two. This leads to calculations involving nets of conics in the plane, invariants of the curves, syzygies, and Hessians. It is emphasized that classical methods, (that is those developed for geometries over fields of zero characteristic), do not always s...

Holomorphic curves omitting five planes in projective space

In 1928 H. Cartan proved an extension of Montel’s normality criterion to holomorphic curves in complex projective plane P2. He also conjectured that a similar result is true for holomorphic curves in P for any n. Recently the author constructed a counterexample to this conjecture for any n ≥ 3. In this paper we show how to modify Cartan’s conjecture so that it becomes true, at least for n = 3.

Dynamics on blowups of the projective plane

Mirror Symmetry for Weighted Projective Planes and Their Noncommutative Deformations