2015/11/11 by Albert, Mario, Bertone, Cristina, Roggero, Margherita +1 · 1 citation
#13D02 #13P10 #14A15 #14C05 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1511.03547
Let \Bbbk be a field of arbitrary characteristic, A a Noetherian \Bbbk-algebra and consider the polynomial ring A[\mathbf x]=A[x0,…,xn]. 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 ⊕m'i=1 A[\mathbf x](-di)). 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.