Computing Quot schemes via marked bases over quasi-stable modules

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