2,180 research outputs found
Deformations of modules of maximal grade and the Hilbert scheme at determinantal schemes
Let R be a polynomial ring and M a finitely generated graded R-module of
maximal grade (which means that the ideal I_t(\cA) generated by the maximal
minors of a homogeneous presentation matrix, \cA, of M has maximal codimension
in R). Suppose X:=Proj(R/I_t(\cA)) is smooth in a sufficiently large open
subset and dim X > 0. Then we prove that the local graded deformation functor
of M is isomorphic to the local Hilbert (scheme) functor at X \subset Proj(R)
under a week assumption which holds if dim X > 1. Under this assumptions we get
that the Hilbert scheme is smooth at (X), and we give an explicit formula for
the dimension of its local ring. As a corollary we prove a conjecture of R. M.
Mir\'o-Roig and the author that the closure of the locus of standard
determinantal schemes with fixed degrees of the entries in a presentation
matrix is a generically smooth component V of the Hilbert scheme. Also their
conjecture on the dimension of V is proved for dim X > 0. The cohomology
H^i_{*}({\cN}_X) of the normal sheaf of X in Proj(R) is shown to vanish for 0 <
i < dim X-1. Finally the mentioned results, slightly adapted, remain true
replacing R by any Cohen-Macaulay quotient of a polynomial ring.Comment: 24 page
A Language Description is More than a Metamodel
Within the context of (software) language engineering, language descriptions are considered first class citizens. One of the ways to describe languages is by means of a metamodel, which represents the abstract syntax of the language. Unfortunately, in this process many language engineers forget the fact that a language also needs a concrete syntax and a semantics. In this paper I argue that neither of these can be discarded from a language description. In a good language description the abstract syntax is the central element, which functions as pivot between concrete syntax and semantics. Furthermore, both concrete syntax and semantics should be described in a well-defined formalism
Families of Artinian and one-dimensional algebras
The purpose of this paper is to study families of Artinian or one dimensional
quotients of a polynomial ring with a special look to level algebras. Let
\GradAlg^H(R) be the scheme parametrizing graded quotients of with
Hilbert function . Let be any graded surjection of quotients of
with Hilbert function and , and h-vectors
and , respectively. If \depth A = \dim A \leq
1 and is a ``truncation'' of in the sense that
for some , then we
show there is a close relationship between \GradAlg^{H_A}(R) and
\GradAlg^{H_B}(R) concerning e.g. smoothness and dimension at the points
and respectively, provided is a complete intersection or
provided the Castelnuovo-Mumford regularity of is at least 3 (sometimes 2)
larger than the regularity of . In the complete intersection case we
generalize this relationship to ``non-truncated'' Artinian algebras which
are compressed or close to being compressed. For more general Artinian algebras
we describe the dual of the tangent and obstruction space of deformations in a
manageable form which we make rather explicit for level algebras of
Cohen-Macaulay type 2. This description and a linkage theorem for families
allow us to prove a conjecture of Iarrobino on the existence of at least two
irreducible components of \GradAlg^H(R), , whose
general elements are Artinian level algebras of type 2.Comment: 29 page
Families of artinian and low dimensional determinantal rings
Let GradAlg(H) be the scheme parameterizing graded quotients of
R=k[x_0,...,x_n] with Hilbert function H (it is a subscheme of the Hilbert
scheme of P^n if we restrict to quotients of positive dimension, see definition
below). A graded quotient A=R/I of codimension c is called standard
determinantal if the ideal I can be generated by the t by t minors of a
homogeneous t by (t+c-1) matrix (f_{ij}). Given integers a_0\le a_1\le ...\le
a_{t+c-2} and b_1\le ...\le b_t, we denote by W_s(\underline{b};\underline{a})
the stratum of GradAlg(H) of determinantal rings where f_{ij} \in R are
homogeneous of degrees a_j-b_i.
In this paper we extend previous results on the dimension and codimension of
W_s(\underline{b};\underline{a}) in GradAlg(H) to {\it artinian determinantal
rings}, and we show that GradAlg(H) is generically smooth along
W_s(\underline{b};\underline{a}) under some assumptions. For zero and one
dimensional determinantal schemes we generalize earlier results on these
questions. As a consequence we get that the general element of a component W of
the Hilbert scheme of P^n is glicci provided W contains a standard
determinantal scheme satisfying some conditions. We also show how certain ghost
terms disappear under deformation while other ghost terms remain and are
present in the minimal resolution of a general element of GradAlg(H).Comment: Postprint replacing preprint. 29 pages. Online 26.May 2017 in Journal
of Pure and Applied Algebr
Connecting the Dots: Towards Continuous Time Hamiltonian Monte Carlo
Continuous time Hamiltonian Monte Carlo is introduced, as a powerful
alternative to Markov chain Monte Carlo methods for continuous target
distributions. The method is constructed in two steps: First Hamiltonian
dynamics are chosen as the deterministic dynamics in a continuous time
piecewise deterministic Markov process. Under very mild restrictions, such a
process will have the desired target distribution as an invariant distribution.
Secondly, the numerical implementation of such processes, based on adaptive
numerical integration of second order ordinary differential equations is
considered. The numerical implementation yields an approximate, yet highly
robust algorithm that, unlike conventional Hamiltonian Monte Carlo, enables the
exploitation of the complete Hamiltonian trajectories (hence the title). The
proposed algorithm may yield large speedups and improvements in stability
relative to relevant benchmarks, while incurring numerical errors that are
negligible relative to the overall Monte Carlo errors
The Hilbert Scheme of Buchsbaum space curves
We consider the Hilbert scheme H(d,g) of space curves C with homogeneous
ideal I(C):=H_{*}^0(\sI_C) and Rao module M:=H_{*}^1(\sI_C). By taking suitable
generizations (deformations to a more general curve) C' of C, we simplify the
minimal free resolution of I(C) by e.g. making consecutive free summands
(ghost-terms) disappear in a free resolution of I(C'). Using this for Buchsbaum
curves of diameter one (M_v \ne 0 for only one v), we establish a one-to-one
correspondence between the set \sS of irreducible components of H(d,g) that
contain (C) and a set of minimal 5-tuples that specializes in an explicit
manner to a 5-tuple of certain graded Betti numbers of C related to
ghost-terms. Moreover we almost completely (resp. completely) determine the
graded Betti numbers of all generizations of C (resp. all generic curves of
\sS), and we give a specific description of the singular locus of the Hilbert
scheme of curves of diameter at most one. We also prove some semi-continuity
results for the graded Betti numbers of any space curve under some assumptions.Comment: Minor changes in Thm. 6.1 where the particular case (v) is corrected
(this inaccuracy occurs also in the published version in Annales de
l'institut Fourier, 2012); 23 page
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