3,576 research outputs found
A note on the multiplicity of determinantal ideals
Herzog, Huneke, and Srinivasan have conjectured that for any homogeneous
-algebra, the multiplicity is bounded above by a function of the maximal
degrees of the syzygies and below by a function of the minimal degrees of the
syzygies. The goal of this paper is to establish the multiplicity conjecture of
Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay
algebras over a field for -algebras being a
determinantal ideal of arbitrary codimension
Ideals of general forms and the ubiquity of the Weak Lefschetz property
Let be positive integers and let be an
ideal generated by general forms of degrees , respectively, in a
polynomial ring with variables. When all the degrees are the same we
give a result that says, roughly, that they have as few first syzygies as
possible. In the general case, the Hilbert function of has been
conjectured by Fr\"oberg. In a previous work the authors showed that in many
situations the minimal free resolution of must have redundant terms which
are not forced by Koszul (first or higher) syzygies among the (and hence
could not be predicted from the Hilbert function), but the only examples came
when . Our second main set of results in this paper show that further
examples can be obtained when . We also show that if
Fr\"oberg's conjecture on the Hilbert function is true then any such redundant
terms in the minimal free resolution must occur in the top two possible degrees
of the free module. Related to the Fr\"oberg conjecture is the notion of Weak
Lefschetz property. We continue the description of the ubiquity of this
property. We show that any ideal of general forms in has
it. Then we show that for certain choices of degrees, any complete intersection
has it and any almost complete intersection has it. Finally, we show that most
of the time Artinian ``hypersurface sections'' of zeroschemes have it.Comment: 24 page
On the intersection of ACM curves in \PP^3
Bezout's theorem gives us the degree of intersection of two properly
intersecting projective varieties. As two curves in P^3 never intersect
properly, Bezout's theorem cannot be directly used to bound the number of
intersection points of such curves. In this work, we bound the maximum number
of intersection points of two integral ACM curves in P^3. The bound that we
give is in many cases optimal as a function of only the degrees and the initial
degrees of the curves
Cohomological characterization of vector bundles on multiprojective spaces
We show that Horrock's criterion for the splitting of vector bundles on
\PP^n can be extended to vector bundles on multiprojective spaces and to
smooth projective varieties with the weak CM property (see Definition 3.11). As
a main tool we use the theory of -blocks and Beilinson's type spectral
sequences. Cohomological characterizations of vector bundles are also showed
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