Expand all Collapse all | Results 1 - 25 of 35 |
1. CJM Online first
Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with $\mu=2$ |
Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with $\mu=2$ We exhibit a set of minimal generators of the defining ideal of the
Rees Algebra associated with the ideal of three bivariate homogeneous
polynomials parametrizing a proper rational curve in projective plane,
having a minimal syzygy of degree 2.
Keywords:Rees Algebras, rational plane curves, minimal generators Categories:13A30, 14H50 |
2. CJM 2012 (vol 66 pp. 3)
On Hilbert Covariants Let $F$ denote a binary form of order $d$ over the
complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant
$\mathcal{H}_{r,d}(F)$ vanishes exactly when $F$ is the perfect power of an
order $r$ form. In geometric terms, the coefficients of $\mathcal{H}$ give
defining equations for the image variety $X$ of an embedding $\mathbf{P}^r
\hookrightarrow \mathbf{P}^d$. In this paper we describe a new construction of
the Hilbert covariant; and simultaneously situate it into a wider class of
covariants called the GÃ¶ttingen covariants, all of which vanish on
$X$. We prove that the ideal generated by the coefficients of $\mathcal{H}$
defines $X$ as a scheme. Finally, we exhibit a generalisation of the
GÃ¶ttingen covariants to $n$-ary forms using the classical Clebsch transfer principle.
Keywords:binary forms, covariants, $SL_2$-representations Categories:14L30, 13A50 |
3. CJM 2012 (vol 65 pp. 823)
Symbolic Powers Versus Regular Powers of Ideals of General Points in $\mathbb{P}^1 \times \mathbb{P}^1$ |
Symbolic Powers Versus Regular Powers of Ideals of General Points in $\mathbb{P}^1 \times \mathbb{P}^1$ Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke
raised the problem of which symbolic powers of an ideal
are contained in a given ordinary power of the ideal.
Bocci-Harbourne developed methods to address this problem,
which involve asymptotic numerical characters of
symbolic powers of the ideals. Most of the work
done up to now has been done for ideals defining 0-dimensional
subschemes of projective space.
Here we focus on certain subschemes given by
a union of lines in $\mathbb{P}^3$ which can also be viewed
as points in $\mathbb{P}^1 \times \mathbb{P}^1$.
We also obtain results on the
closely related problem, studied by Hochster and by Li-Swanson, of
determining situations for which
each symbolic power of an ideal is an ordinary power.
Keywords:symbolic powers, multigraded, points Categories:13F20, 13A15, 14C20 |
4. CJM 2012 (vol 65 pp. 634)
Laplace Equations and the Weak Lefschetz Property We prove that $r$ independent homogeneous polynomials of the same degree $d$
become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety
whose $(d-1)$-osculating spaces have dimension smaller than expected. This gives an equivalence
between an algebraic notion (called Weak Lefschetz Property)
and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case,
some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture.
Keywords:osculating space, weak Lefschetz property, Laplace equations, toric threefold Categories:13E10, 14M25, 14N05, 14N15, 53A20 |
5. CJM 2010 (vol 62 pp. 1131)
Moduli Spaces of Reflexive Sheaves of Rank 2
Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of
dimension at least two and let $X \subset Y$ be the zero set of a section
$\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the
functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform
$\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two
forgetful maps between the functors, we prove that the scheme structure of
\emph{e.g.,} the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$
at $(\mathcal{F})$, and the scheme structure at $(X)$ of the Hilbert scheme of curves
on $Y$ become closely related. Using this relationship, we get criteria for the
dimension and smoothness of $ {\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming $
{\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose
deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies $ {_{0}\! \textrm{Ext}^2}(M ,M ) = 0 $
(\emph{e.g.,} of diameter at most 2),
we get necessary and sufficient conditions of unobstructedness that coincide
in the diameter one case. The conditions are further equivalent to the
vanishing of certain graded Betti numbers of the free graded minimal
resolution of $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible
component of ${\rm M}_{\mathbb{P}^{3}}(P)$ containing a reflexive sheaf of diameter
one is reduced (generically smooth) and we compute its dimension. We also
determine a good lower bound for the dimension of any component of ${\rm
M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with ``small''
deficiency module $M$.
Keywords:moduli space, reflexive sheaf, Hilbert scheme, space curve, Buchsbaum sheaf, unobstructedness, cup product, graded Betti numbers.xdvi Categories:14C05, qqqqq14D22, 14F05, 14J10, 14H50, 14B10, 13D02, 13D07 |
6. CJM 2009 (vol 62 pp. 721)
Formal Fibers of Unique Factorization Domains Let $(T,M)$ be a complete local (Noetherian) ring such that $\dim T\geq 2$ and
$|T|=|T/M|$ and let $\{p_i\} _{i \in \mathcal I}$ be a collection of
elements of T indexed by a set $\mathcal I$ so that $|\mathcal I | < |T|$.
For each $i \in \mathcal{I}$, let $C_i$:={$Q_{i1}$,$\dots$,$Q_{in_i}$}
be a set of nonmaximal prime ideals containing $p_i$ such that the $Q_{ij}$
are incomparable and $p_i\in Q_{jk}$ if and only if $i=j$. We provide necessary
and sufficient conditions so that T is the ${\bf m}$-adic completion of a local unique
factorization domain $(A, {\bf m})$, and for each $i \in \mathcal I$, there exists a unit
$t_i$ of T so that $p_{i}t_i \in A$ and $C_i$
is the set of prime ideals $Q$ of $T$ that are maximal with respect to the condition
that $Q \cap A = p_{i}t_{i}A$.
We then use this result to construct a
(nonexcellent) unique factorization domain containing many ideals for which tight closure and
completion do not commute. As another application, we construct a unique factorization
domain A most of whose formal fibers are geometrically regular.
Categories:13J10, 13J05 |
7. CJM 2009 (vol 61 pp. 888)
Face Ring Multiplicity via CM-Connectivity Sequences The multiplicity conjecture of Herzog, Huneke, and Srinivasan
is verified for the face rings of the following classes of
simplicial complexes: matroid complexes, complexes of dimension
one and two,
and Gorenstein complexes of dimension at most four.
The lower bound part of this conjecture is also established for the
face rings of all doubly Cohen--Macaulay complexes whose 1-skeleton's
connectivity does not exceed the codimension plus one as well as for
all $(d-1)$-dimensional $d$-Cohen--Macaulay complexes.
The main ingredient of the proofs is a new interpretation
of the minimal shifts in the resolution of the face ring
$\field[\Delta]$ via the Cohen--Macaulay connectivity of the
skeletons of $\Delta$.
Categories:13F55, 52B05;, 13H15;, 13D02;, 05B35 |
8. CJM 2009 (vol 61 pp. 950)
Infinitesimal Invariants in a Function Algebra Let $G$ be a reductive connected linear algebraic group
over an algebraically closed field of positive
characteristic and let $\g$ be its Lie algebra.
First we extend a well-known result about the Picard group of a
semi-simple group to reductive groups.
Then we prove that if the derived group is simply connected
and $\g$ satisfies a
mild condition, the algebra $K[G]^\g$ of regular functions
on $G$ that are invariant under the action of $\g$ derived
from the conjugation action is a unique factorisation domain.
Categories:20G15, 13F15 |
9. CJM 2009 (vol 61 pp. 930)
Prolongations and Computational Algebra We explore the geometric notion of prolongations in the setting of
computational algebra, extending results of Landsberg and Manivel
which relate prolongations to equations for secant varieties. We also
develop methods for computing prolongations that are combinatorial in
nature. As an application, we use prolongations to derive a new
family of secant equations for the binary symmetric model in
phylogenetics.
Categories:13P10, 14M99 |
10. CJM 2009 (vol 61 pp. 762)
The Hilbert Coefficients of the Fiber Cone and the $a$-Invariant of the Associated Graded Ring Let $(A,\m)$ be a Noetherian local ring with infinite residue
field and let $I$ be an ideal in $A$ and let $F(I) =
\bigoplus_{n \geq 0}I^n/\m I^n$ be the fiber cone of $I$.
We prove certain relations among the Hilbert coefficients $f_0(I),f_1(I), f_2(I)$ of $F(I)$
when the $a$-invariant of the associated graded ring $G(I)$ is negative.
Keywords:fiber cone, $a$-invariant, Hilbert coefficients of fiber cone Categories:13A30, 13D40 |
11. CJM 2009 (vol 61 pp. 76)
Ascent Properties of Auslander Categories Let $R$ be a homomorphic image of a Gorenstein local ring. Recent
work has shown that there is a bridge between Auslander categories
and modules of finite Gorenstein homological dimensions over $R$.
We use Gorenstein dimensions to prove new results about Auslander
categories and vice versa. For example, we establish base change
relations between the Auslander categories of the source and target
rings of a homomorphism $\varphi \colon R \to S$ of finite flat dimension.
Keywords:Auslander categories, Gorenstein dimensions, ascent properties, Auslander--Buchsbaum formulas Categories:13D05, 13D07, 13D25 |
12. CJM 2009 (vol 61 pp. 205)
Representations of Non-Negative Polynomials, Degree Bounds and Applications to Optimization Natural sufficient conditions for a polynomial to have a local minimum
at a point are considered. These conditions tend to hold with
probability $1$. It is shown that polynomials satisfying these
conditions at each minimum point have nice presentations in terms of
sums of squares. Applications are given to optimization on a compact
set and also to global optimization. In many cases, there are degree
bounds for such presentations. These bounds are of theoretical
interest, but they appear to be too large to be of much practical use
at present. In the final section, other more concrete degree bounds
are obtained which ensure at least that the feasible set of solutions
is not empty.
Categories:13J30, 12Y05, 13P99, 14P10, 90C22 |
13. CJM 2009 (vol 61 pp. 29)
The Minimal Resolution Conjecture for Points on the Cubic Surface In this paper we prove that a generalized version of the Minimal
Resolution Conjecture given by Musta\c{t}\v{a} holds for certain
general sets of points on a smooth cubic surface $X \subset
\PP^3$. The main tool used is Gorenstein liaison theory and, more
precisely, the relationship between the free resolutions of two linked schemes.
Categories:13D02, 13C40, 14M05, 14M07 |
14. CJM 2008 (vol 60 pp. 721)
Uniform Linear Bound in Chevalley's Lemma We obtain a uniform linear bound for the Chevalley function at a point in
the source of an analytic mapping that is regular in the sense of
Gabrielov. There is a version of
Chevalley's lemma also along a fibre, or at a point of the image of a proper
analytic mapping. We get a uniform linear bound for the Chevalley
function of a closed Nash (or formally Nash) subanalytic set.
Keywords:Chevalley function, regular mapping, Nash subanalytic set Categories:13J07, 32B20, 13J10, 32S10 |
15. CJM 2008 (vol 60 pp. 556)
Polarization of Separating Invariants We prove a characteristic free version of Weyl's theorem on
polarization. Our result is an exact analogue of Weyl's theorem, the
difference being that our statement is about separating invariants
rather than generating invariants. For the special case of finite
group actions we introduce the concept of \emph{cheap polarization},
and show that it is enough to take cheap polarizations of invariants
of just one copy of a representation to obtain separating vector
invariants for any number of copies. This leads to upper bounds on
the number and degrees of separating vector invariants of finite
groups.
Keywords:Jan Draisma, Gregor Kemper, David Wehlau Categories:13A50, 14L24 |
16. CJM 2008 (vol 60 pp. 391)
The Geometry of the Weak Lefschetz Property and Level Sets of Points In a recent paper, F. Zanello showed that level Artinian algebras in 3
variables can fail to have the Weak Lefschetz Property (WLP), and can
even fail to have unimodal Hilbert function. We show that the same is
true for the Artinian reduction of reduced, level sets of points in
projective 3-space. Our main goal is to begin an understanding of how
the geometry of a set of points can prevent its Artinian reduction
from having WLP, which in itself is a very algebraic notion. More
precisely, we produce level sets of points whose Artinian reductions
have socle types 3 and 4 and arbitrary socle degree $\geq 12$ (in the
worst case), but fail to have WLP. We also produce a level set of
points whose Artinian reduction fails to have unimodal Hilbert
function; our example is based on Zanello's example. Finally, we show
that a level set of points can have Artinian reduction that has WLP
but fails to have the Strong Lefschetz Property. While our
constructions are all based on basic double G-linkage, the
implementations use very different methods.
Keywords:Weak Lefschetz Property, Strong Lefschetz Property, basic double G-linkage, level, arithmetically Gorenstein, arithmetically Cohen--Macaulay, socle type, socle degree, Artinian reduction Categories:13D40, 13D02, 14C20, 13C40, 13C13, 14M05 |
17. CJM 2007 (vol 59 pp. 880)
Radical Ideals in Valuation Domains An ideal $I$ of a ring $R$ is called a radical ideal if
$I={\mathcalR}(R)$ where ${\mathcal R}$ is a radical in the sense of
Kurosh--Amitsur. The main theorem of this paper asserts that if $R$
is a valuation domain, then a proper ideal $I$ of $R$ is a radical
ideal if and only if $I$ is a distinguished ideal of $R$ (the
latter property means that if $J$ and $K$ are ideals of $R$ such
that $J\subset I\subset K$ then we cannot have $I/J\cong K/I$ as
rings) and that such an ideal is necessarily prime. Examples are
exhibited which show that, unlike prime ideals, distinguished
ideals are not characterizable in terms of a property of the
underlying value group of the valuation domain.
Categories:16N80, 13A18 |
18. CJM 2007 (vol 59 pp. 109)
On Fiber Cones of $\m$-Primary Ideals Two formulas for the multiplicity of the fiber cone
$F(I)=\bigoplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$-primary ideal of
a $d$-dimensional Cohen--Macaulay local ring $(R,\m)$ are derived in
terms of the mixed multiplicity $e_{d-1}(\m | I)$, the multiplicity
$e(I)$, and superficial elements. As a consequence, the
Cohen--Macaulay property of $F(I)$ when $I$ has minimal mixed
multiplicity or almost minimal mixed multiplicity is characterized
in terms of the reduction number of $I$ and lengths of certain ideals.
We also characterize the Cohen--Macaulay and Gorenstein properties of
fiber cones of $\m$-primary ideals with a $d$-generated minimal
reduction $J$ satisfying $\ell(I^2/JI)=1$ or
$\ell(I\m/J\m)=1.$
Keywords:fiber cones, mixed multiplicities, joint reductions, Cohen--Macaulay fiber cones, Gorenstein fiber cones, ideals having minimal and almost minimal mixed multiplicities Categories:13H10, 13H15, 13A30, 13C15, 13A02 |
19. CJM 2005 (vol 57 pp. 1178)
Asymptotic Behavior of the Length of Local Cohomology Let $k$ be a field of characteristic 0, $R=k[x_1, \ldots, x_d]$ be a polynomial ring,
and $\mm$ its maximal homogeneous ideal. Let $I \subset R$ be a homogeneous ideal in
$R$. Let $\lambda(M)$ denote the length of an $R$-module $M$. In this paper, we show
that
$$
\lim_{n \to \infty} \frac{\l\bigl(H^0_{\mathfrak{m}}(R/I^n)\bigr)}{n^d}
=\lim_{n \to \infty} \frac{\l\bigl(\Ext^d_R\bigl(R/I^n,R(-d)\bigr)\bigr)}{n^d}
$$
always exists. This limit has been shown to be ${e(I)}/{d!}$ for $m$-primary ideals
$I$ in a local Cohen--Macaulay ring, where $e(I)$ denotes the multiplicity
of $I$. But we find that this limit may not be rational in general. We give an example
for which the limit is an irrational number thereby showing that the lengths of these
extention modules may not have polynomial growth.
Keywords:powers of ideals, local cohomology, Hilbert function, linear growth Categories:13D40, 14B15, 13D45 |
20. CJM 2005 (vol 57 pp. 724)
Some Results on Surfaces of General Type In this article we prove some new results on projective normality, normal
presentation and higher syzygies for surfaces of general type, not
necessarily smooth, embedded by adjoint linear series. Some of the
corollaries of more general results include: results on property $N_p$
associated to $K_S \otimes B^{\otimes n}$ where $B$ is base-point free and
ample divisor with $B\otimes K^*$ {\it nef}, results for pluricanonical
linear systems and results giving effective bounds for adjoint linear series
associated to ample bundles. Examples in the last section show that the results
are optimal.
Categories:13D02, 14C20, 14J29 |
21. CJM 2005 (vol 57 pp. 400)
Generalized $k$-Configurations In this paper, we find configurations of points in $n$-dimensional
projective space ($\proj ^n$) which simultaneously generalize both
$k$-configurations and reduced 0-dimensional complete intersections.
Recall that $k$-configurations in $\proj ^2$ are disjoint unions of
distinct points on lines and in $\proj ^n$ are inductively disjoint
unions of $k$-configurations on hyperplanes, subject to certain
conditions. Furthermore, the Hilbert function of a $k$-configuration
is determined from those of the smaller $k$-configurations. We call
our generalized constructions $k_D$-configurations, where $D=\{ d_1,
\ldots ,d_r\}$ (a set of $r$ positive integers with repetition
allowed) is the type of a given complete intersection in $\proj ^n$.
We show that the Hilbert function of any $k_D$-configuration can be
obtained from those of smaller $k_D$-configurations. We then provide
applications of this result in two different directions, both of which
are motivated by corresponding results about $k$-configurations.
Categories:13D40, 14M10 |
22. CJM 2004 (vol 56 pp. 742)
Similarity Classification of Cowen-Douglas Operators Let $\cal H$ be a complex separable Hilbert space
and ${\cal L}({\cal H})$ denote the collection of
bounded linear operators on ${\cal H}$.
An operator $A$ in ${\cal L}({\cal H})$
is said to be strongly irreducible, if
${\cal A}^{\prime}(T)$, the commutant of $A$, has no non-trivial idempotent.
An operator $A$ in ${\cal L}({\cal H})$ is said to a Cowen-Douglas
operator, if there exists $\Omega$, a connected open subset of
$C$, and $n$, a positive integer, such that
(a) ${\Omega}{\subset}{\sigma}(A)=\{z{\in}C; A-z {\text {not invertible}}\};$
(b) $\ran(A-z)={\cal H}$, for $z$ in $\Omega$;
(c) $\bigvee_{z{\in}{\Omega}}$\ker$(A-z)={\cal H}$ and
(d) $\dim \ker(A-z)=n$ for $z$ in $\Omega$.
In the paper, we give a similarity classification of strongly
irreducible Cowen-Douglas operators by using the $K_0$-group of
the commutant algebra as an invariant.
Categories:47A15, 47C15, 13E05, 13F05 |
23. CJM 2004 (vol 56 pp. 716)
Fat Points in $\mathbb{P}^1 \times \mathbb{P}^1$ and Their Hilbert Functions We study the Hilbert functions of fat points in $\popo$.
If $Z \subseteq \popo$ is an arbitrary fat point scheme, then
it can be shown that for every $i$ and $j$ the values of the Hilbert
function $_{Z}(l,j)$ and $H_{Z}(i,l)$ eventually become constant for
$l \gg 0$. We show how to determine these eventual values
by using only the multiplicities of the points, and the
relative positions of the points in $\popo$. This enables
us to compute all but a finite number values of $H_{Z}$
without using the coordinates of points.
We also characterize the ACM fat point schemes
sing our description of the eventual behaviour. In fact,
n the case that $Z \subseteq \popo$ is ACM, then
the entire Hilbert function and its minimal free resolution
depend solely on knowing the eventual values of the Hilbert function.
Keywords:Hilbert function, points, fat points, Cohen-Macaulay, multi-projective space Categories:13D40, 13D02, 13H10, 14A15 |
24. CJM 2003 (vol 55 pp. 1019)
More Eventual Positivity for Analytic Functions Eventual positivity problems for real convergent Maclaurin series lead
to density questions for sets of harmonic functions. These are solved
for large classes of series, and in so doing, asymptotic estimates are
obtained for the values of the series near the radius of convergence
and for the coefficients of convolution powers.
Categories:30B10, 30D15, 30C50, 13A99, 41A58, 42A16 |
25. CJM 2003 (vol 55 pp. 711)
Adic Topologies for the Rational Integers A topology on $\mathbb{Z}$, which gives a nice proof that the
set of prime integers is infinite, is characterised and examined.
It is found to be homeomorphic to $\mathbb{Q}$, with a compact
completion homeomorphic to the Cantor set. It has a natural place
in a family of topologies on $\mathbb{Z}$, which includes the
$p$-adics, and one in which the set of rational primes $\mathbb{P}$
is dense. Examples from number theory are given, including the
primes and squares, Fermat numbers, Fibonacci numbers and $k$-free
numbers.
Keywords:$p$-adic, metrizable, quasi-valuation, topological ring,, completion, inverse limit, diophantine equation, prime integers,, Fermat numbers, Fibonacci numbers Categories:11B05, 11B25, 11B50, 13J10, 13B35 |