A decomposition of the Hilbert scheme given by Gröbner schemes

We consider the Hilbert scheme H which is the scheme parameterizing all closed subschemes of the projective space P^n with Hilbert polynomial P. If we fix a monomial order < on the polynomial ring S with n+1 variables, each homogeneous ideal in S has a unique reduced Grobner basis with respect to <. Using this fact we can decompose the Hilbert scheme H into locally closed subschemes of H called the Grobner schemes. On the other hand, Bialynicki-Birula shows that any smooth projective scheme with a 1-dimensional torus action has a cell decomposition called the Bialynicki-Birula decomposition.

In this talk, I would like to explain Gröbner schemes and the decomposition. I introduce a 1-dimensional torus action on the Hilbert scheme H which is compatible with < and I show that the decomposition given by the Gröbner schemes can be constructed by such torus action in the sense of Bialynicki-Birula.