Ruled cubic surfaces in PG(4,q), Baer subplanes of PG(2,q2) and Hermitian curves

Baer Subplanes, Ovals and Unitals

Rational curves and ruled orders on surfaces

We study ruled orders. These arise naturally in the Mori program for orders on projective surfaces and morally speaking are orders on a ruled surface ramified on a bisection and possibly some fibres. We describe fibres of a ruled order and show they are in some sense rational. We also determine the Hilbert scheme of rational curves and hence the corresponding non-commutative Mori contraction. T...

Cubic Ruled Surfaces with Constant Distribution Parameter in E4

A first order invariant of ruled surfaces of E3 is the socalled distribution parameter d in a generator. It is defined as the limit of the quotient of the distance and the angle of the generator and its neighbour. Ruled surfaces with constant parameter of distribution are of special interest and have been studied by many authors. H. Brauner could prove that the only nontrivial cubic ruled surfa...

Algebraic curves, rich points, and doubly-ruled surfaces

We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let k be a field and let L be a collection of n space curves in k, with n << (char(k)) or char(k) = 0. Then either A) there are at most O(n) points in k hit by at least two curves, or B) at least Ω(n) curves from L must lie on a bounded-degree surface, and many of ...

Counting Curves of Any Genus on Rational Ruled Surfaces

In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface Fn. We compute the number of such curves through the appropriate number of fixed general points on Fn (Theorem 1.1), and the number of such curves which are irreducible (Theorem 1.3). These numbers are known as Severi degrees; they are the degrees of unions of components of the Hilbert...