Computing Quot schemes via marked bases over quasi-stable modules

Abstract

Let $ \Bbbk$ be a field of arbitrary characteristic, $A$ a Noetherian $ \Bbbk$-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 Gr\"obner 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 $\oplus^{m'}_{i=1} A[\mathbf x](-d_i)$). 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.