19 research outputs found
Generic initial ideals of points and curves
Let I be the defining ideal of a smooth irreducible complete intersection
space curve C with defining equations of degrees a and b. We use the partial
elimination ideals introduced by Mark Green to show that the lexicographic
generic initial ideal of I has Castelnuovo-Mumford regularity 1+ab(a-1)(b-1)/2
with the exception of the case a=b=2, where the regularity is 4. Note that
ab(a-1)(b-1)/2 is exactly the number of singular points of a general projection
of C to the plane. Additionally, we show that for any term ordering tau, the
generic initial ideal of a generic set of points in P^r is a tau-segment ideal.Comment: AMS Latex, 20 pages, v2: 19 pages, updated references, minor
corrections, proofs of some elementary facts shortened or removed. To appear
in J. Symbolic Computatio
Generic circuits sets and general initial ideals with respect to weights
We study the set of circuits of a homogeneous ideal and that of its
truncations, and introduce the notion of generic circuits set. We show how this
is a well-defined invariant that can be used, in the case of initial ideals
with respect to weights, as a counterpart of the (usual) generic initial ideal
with respect to monomial orders. As an application we recover the existence of
the generic fan introduced by R\"omer and Schmitz for studying generic tropical
varieties. We also consider general initial ideals with respect to weights and
show, in analogy to the fact that generic initial ideals are Borel-fixed, that
these are fixed under the action of certain Borel subgroups of the general
linear group.Comment: 10 page
The Degree Complexity of Smooth Surfaces of codimension 2
D.Bayer and D.Mumford introduced the degree complexity of a projective scheme
for the given term order as the maximal degree of the reduced Gr\"{o}bner
basis. It is well-known that the degree complexity with respect to the graded
reverse lexicographic order is equal to the Castelnuovo-Mumford regularity
(\cite{BS}). However, little is known about the degree complexity with respect
to the graded lexicographic order (\cite{A}, \cite{CS}).
In this paper, we study the degree complexity of a smooth irreducible surface
in \p^4 with respect to the graded lexicographic order and its geometric
meaning. Interestingly, this complexity is closely related to the invariants of
the double curve of a surface under the generic projection.
As results, we prove that except a few cases, the degree complexity of a
smooth surface of degree with in \p^4 is
given by , where is a
double curve of degree under a generic
projection of (Theorem \ref{mainthm2}). Exceptional cases are either a
rational normal scroll or a complete intersection surface of -type or a
Castelnuovo surface of degree 5 in \p^4 whose degree complexities are in fact
equal to their degrees. This complexity can also be expressed only in terms of
the maximal degree of defining equations of (Corollary \ref{cor:01} and
\ref{cor:02}).
We also provide some illuminating examples of our results via calculations
done with {\it Macaulay 2} (Example \ref{Exam:01}).Comment: 18 pages. Some theorems and examples are added. The case of singular
space curves is delete
Partial elimination ideals and secant cones
For any k \in \Nat, we show that the cone of -secant lines of a
closed subscheme over an algebraically closed field
running through a closed point is defined by the
-th partial elimination ideal of with respect to . We use this fact
to give an algorithm for computing secant cones. Also, we show that under
certain conditions partial elimination ideals describe the length of the fibres
of a multiple projection in a way similar to the way they do for simple
projections. Finally, we study some examples illustrating these results,
computed by means of {\sc Singular}.Comment: 18 pages; revised version, to appear in Journal of Algebr
Ideals with an assigned initial ideal
The stratum St(J,<) (the homogeneous stratum Sth(J,<) respectively) of a
monomial ideal J in a polynomial ring R is the family of all (homogeneous)
ideals of R whose initial ideal with respect to the term order < is J. St(J,<)
and Sth(J,<) have a natural structure of affine schemes. Moreover they are
homogeneous w.r.t. a non-standard grading called level. This property allows us
to draw consequences that are interesting from both a theoretical and a
computational point of view. For instance a smooth stratum is always isomorphic
to an affine space (Corollary 3.6). As applications, in Sec. 5 we prove that
strata and homogeneous strata w.r.t. any term ordering < of every saturated
Lex-segment ideal J are smooth. For Sth(J,Lex) we also give a formula for the
dimension. In the same way in Sec. 6 we consider any ideal R in k[x0,..., xn]
generated by a saturated RevLex-segment ideal in k[x,y,z]. We also prove that
Sth(R,RevLex) is smooth and give a formula for its dimension.Comment: 14 pages, improved version, some more example
Matrix-F5 algorithms over finite-precision complete discrete valuation fields
Let be a sequence
of homogeneous polynomials with -adic coefficients. Such system may happen,
for example, in arithmetic geometry. Yet, since is not an
effective field, classical algorithm does not apply.We provide a definition for
an approximate Gr{\"o}bner basis with respect to a monomial order We
design a strategy to compute such a basis, when precision is enough and under
the assumption that the input sequence is regular and the ideals are weakly--ideals. The conjecture of Moreno-Socias
states that for the grevlex ordering, such sequences are generic.Two variants
of that strategy are available, depending on whether one lean more on precision
or time-complexity. For the analysis of these algorithms, we study the loss of
precision of the Gauss row-echelon algorithm, and apply it to an adapted
Matrix-F5 algorithm. Numerical examples are provided.Moreover, the fact that
under such hypotheses, Gr{\"o}bner bases can be computed stably has many
applications. Firstly, the mapping sending to the reduced
Gr{\"o}bner basis of the ideal they span is differentiable, and its
differential can be given explicitly. Secondly, these hypotheses allows to
perform lifting on the Grobner bases, from to
or Finally, asking for the same
hypotheses on the highest-degree homogeneous components of the entry
polynomials allows to extend our strategy to the affine case
Segments and Hilbert schemes of points
Using results obtained from the study of homogeneous ideals sharing the same
initial ideal with respect to some term order, we prove the singularity of the
point corresponding to a segment ideal with respect to the revlex term order in
the Hilbert scheme of points in . In this context, we look inside
properties of several types of "segment" ideals that we define and compare.
This study led us to focus our attention also to connections between the shape
of generators of Borel ideals and the related Hilbert polynomial, providing an
algorithm for computing all saturated Borel ideals with the given Hilbert
polynomial.Comment: 19 pages, 2 figures. Comments and suggestions are welcome