An explicit construction of ruled surfaces

Abstract:

In this talk we describe a general algorithm to  compute,
via computer algebra systems,
an explicit set of generators of the ideals of the projective embeddings
of ruled surfaces, the projectivization of rank two vector bundles
over curves, such that  the  fibres are embedded as  smooth rational curves.

There are two different applications of our algorithm.
Firstly, when the existence of the ruled surface is known,
an explicit description of the ideal allows us to compute,
besides all the syzygies,
some important algebraic invariants of the surface,
which are not always easy to compute by general formulae
or by geometric arguments, for instance the k-regularity.
Secondly, it is possible to prove the existence
of new examples of smooth surfaces.

The method can be implemented over any computer algebra system
able to deal with commutative algebra
and Groebner-basis computations.

According to time, an implementation of our algorithms for the computer
algebra system Macaulay2 and explicit examples will be exposed.