Computing Quot schemes via marked bases over quasi-stable modules

Abstract

Let $\mathbb{k}$ be a field of arbitrary characteristic, $A$ a Noetherian $\mathbb{k}$-algebra and consider the polynomial ring $A[\mathbf{x}]=A[x_0,\dots ,x_n]$. We consider homogeneous submodules of $A[\mathbf{x}{]}^m$ having a special set of generators: a marked basis over a quasi-stable module. Such a marked basis inherits several good properties of a Gröbner basis, including a Noetherian reduction relation. The set of submodules of $A[\mathbf{x}{]}^m$ having a marked basis over a given quasi-stable module has an affine scheme structure that we are able to exhibit. Furthermore, the syzygies of a module generated by such a marked basis are generated by a marked basis, too (over a suitable quasi-stable module in ${\oplus }_{i=1}^{m^{\prime }}A[\mathbf{x}](-d_i)$). We apply the construction of marked bases and related properties to the investigation of Quot functors (and schemes). More precisely, for a given Hilbert polynomial, we can explicitely construct (up to the action of a general linear group) an open cover of the corresponding Quot functor made up of open functors represented by affine schemes. This gives a new proof that the Quot functor is the functor of points of a scheme. We also exhibit a procedure to obtain the equations defining a given Quot scheme as a subscheme of a suitable Grassmannian. Thanks to the good behaviour of marked bases with respect to Castelnuovo-Mumford regularity, we can adapt our methods in order to study the locus of the Quot scheme given by an upper bound on the regularity of its points.

28 pages, exposition improved. This version contains the results of the previous one, and also the application to Quot schemes