Computation of Hilbert Schemes

Hilbert schemes are a basic topic in Algebraic Geometry [1, 2]. Methods related to Gröbner bases are fundamental here, as Hartshorne’s proof of the connectedness of Hilbert schemes via generic initial ideals demonstrates [3]. For many purposes, the explicit construction of Hilbert schemes is important. Classical approaches lead to prohibitively large equations. Newer ideas like Gröbner strata [4, 5] lead to improvements, but do not provide open covers. Despite all advances, it has remained a great challenge to construct Hilbert schemes even for very small Hilbert polynomials. A novel approach replacing Gröbner bases by J-marked bases [6, 7] was proposed by the group of Margharita Roggero. For every strongly stable ideal J, one obtains here a larger family which corresponds to an open subscheme and which can be described by equations of low degree. We have implemented several of the new algorithms in the computer algebra system COCOALIB [8]. In the talk we are going to explain these algorithms and give some ideas to improve these algorithms. In addition to that we present some first practical experience which we have made during first computations. Furthermore we report some experiences which we have made when we tried to parallelize these algorithms.