Computing Quot schemes via marked bases over quasi-stable modules

Let k be a field of arbitrary characteristic, A a Noetherian k-algebra and consider the polynomial ring A[x]=A[x_0,....x_n]. We consider graded submodules of A[x]^m having a special set of generators, a marked basis over a quasi-stable module. Such a marked basis shares many interesting properties with a Gröbner basis, including the existence of a Noetherian reduction relation. The set of submodules of A[x]^m having a marked basis over a given quasi-stable module possesses a natural affine scheme structure which we will exhibit. Furthermore, the syzygies of a module with such a marked basis are generated by a marked basis, too (over a suitable quasi-stable module in $\oplus_{i=1}^{m'} A[x](-d_i)$ ). We apply marked bases and related properties to the investigation of Quot functors (and schemes). More precisely, for a given Hilbert polynomial, we explicitly construct (up to the action of a general linear group) an open cover of the corresponding Quot functor, made up of open subfunctors represented by affine schemes. This provides 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.