Components of the Hilbert Scheme and Constructions for Space Curves - Macaulay2/Workshop-2015-Boise GitHub Wiki

Components of the Local Hilbert Scheme

We are making extensive use of the package VersalDeformations in order to establish the number of components of the local Hilbert Scheme of some curves in P^3. Our aim is to create a package that will enable us to detect, at least, a lower bound on the number of smooth components. So far, we have obtained a sound comparison with the examples appearing in the documentation of VersalDeformations.

Our Components package still only contains experimental code.

Constructions for curves in projective space

We are working towards an implementation of classical constructions for space curves. For example, let C be a nondegenerate smooth curve in projective 3-space. We want the equations of

• Secant variety to C (in Plucker coordinates),
• Curve in the Grassmannian induced by the tangent lines to C (in Plucker coordinates).
• Curve induced by the osculating planes of C.