46 research outputs found
Multigraded regularity, a*-invariant and the minimal free resolution
In recent years, two different multigraded variants of Castelnuovo-Mumford
regularity have been developed, namely multigraded regularity, defined by the
vanishing of multigraded pieces of local cohomology modules, and the resolution
regularity vector, defined by the multidegrees in a minimal free resolution. In
this paper, we study the relationship between multigraded regularity and the
resolution regularity vector. Our method is to investigate multigraded variants
of the usual a*-invariant. This, in particular, provides an effective approach
to examining the vanishing of multigraded pieces of local cohomology modules
with respect to different graded irrelevant ideals.Comment: Final version to appear in J. Algebra; 24 page
Asymptotic linearity of regularity and a*-invariant of powers of ideals
Let X = Proj R be a projective scheme over a field k, and let I be an ideal
in R generated by forms of the same degree d. Let Y --> X be the blowing up of
X along the subscheme defined by I, and let f: Y --> Z be the projection of Y
given by the divisor dH - E, where E is the exceptional divisor of the blowup
and H is the pullback of a general hyperplane in X. We investigate how the
asymptotic linearity of the regularity and a*-invariant of I^q (for q large) is
related to invariants of fibers of f.Comment: 11 pages, revision: get rid of the condition that R is a polynomial
ring in the last theorem
Box-shaped matrices and the defining ideal of certain blowup surfaces
We study the defining equations of projective embeddings of the blowup of P^2
at a set of {d+1 \choose 2} number of points in generic position. To do this,
we first generalize the notion of a matrix, its ideal of 2x2 minors to that of
a box-shaped matrix. Our work completes previous works of Geramita and
Gimigliano
Minimal free resolutions and asymptotic behavior of multigraded regularity
Let S be a standard N^k-graded polynomial ring over a field. Let I be a
multigraded homogeneous ideal in S and let M be a finitely generated Z^k-graded
S-module. We prove that the resolution regularity, a multigraded variant of
Castelnuovo-Mumford regularity, of I^nM is asymptotically a linear function.
This shows that the well known Z-graded phenomenon carries to multigraded
situation.Comment: Final version to appear in J. Algebra; 18 page
Embedded Associated Primes of Powers of Square-free Monomial Ideals
An ideal I in a Noetherian ring R is normally torsion-free if
Ass(R/I^t)=Ass(R/I) for all natural numbers t. We develop a technique to
inductively study normally torsion-free square-free monomial ideals. In
particular, we show that if a square-free monomial ideal I is minimally not
normally torsion-free then the least power t such that I^t has embedded primes
is bigger than beta_1, where beta_1 is the monomial grade of I, which is equal
to the matching number of the hypergraph H(I) associated to I. If in addition I
fails to have the packing property, then embedded primes of I^t do occur when
t=beta_1 +1. As an application, we investigate how these results relate to a
conjecture of Conforti and Cornu\'ejols.Comment: 15 pages, changes have been made to the title, introduction, and
background material, and an example has been added. To appear in JPA