481 research outputs found
Blowups and fibers of morphisms
Our object of study is a rational map Psi from projective s-1 space to
projective n-1 space defined by homogeneous forms g1,...,gn, of the same degree
d, in the homogeneous coordinate ring R=k[x1,...,xs] of projective s-1 space.
Our goal is to relate properties of Psi, of the homogeneous coordinate ring
A=k[g1,...,gn] of the variety parametrized by Psi, and of the Rees algebra
R[It], the bihomogeneous coordinate ring of the graph of Psi. For a regular map
Psi, for instance, we prove that R[It] satisfies Serre's condition R_i, for
some positive i, if and only if A satisfies R_{i-1} and Psi is birational onto
its image. Thus, in particular, Psi is birational onto its image if and only if
R[It] satisfies R_1. Either condition has implications for the shape of the
core, namely, the core of I is the multiplier ideal of I to the power s and the
core of I equals the maximal homogeneous ideal of R to the power sd-s+1.
Conversely, for s equal to two, either equality for the core implies
birationality. In addition, by means of the generalized rows of the syzygy
matrix of g1,...,gn, we give an explicit method to reduce the non-birational
case to the birational one when s is equal to 2.Comment: A j-multiplicity interpretation has been adde
A study of singularities on rational curves via syzygies
Consider a rational projective curve C of degree d over an algebraically
closed field k. There are n homogeneous forms g_1,...,g_n of degree d in
B=k[x,y] which parameterize C in a birational, base point free, manner. We
study the singularities of C by studying a Hilbert-Burch matrix phi for the row
vector [g_1,...,g_n]. In the "General Lemma" we use the generalized row ideals
of phi to identify the singular points on C, their multiplicities, the number
of branches at each singular point, and the multiplicity of each branch.
Let p be a singular point on the parameterized planar curve C which
corresponds to a generalized zero of phi. In the "Triple Lemma" we give a
matrix phi' whose maximal minors parameterize the closure, in projective
2-space, of the blow-up at p of C in a neighborhood of p. We apply the General
Lemma to phi' in order to learn about the singularities of C in the first
neighborhood of p. If C has even degree d=2c and the multiplicity of C at p is
equal to c, then we apply the Triple Lemma again to learn about the
singularities of C in the second neighborhood of p.
Consider rational plane curves C of even degree d=2c. We classify curves
according to the configuration of multiplicity c singularities on or infinitely
near C. There are 7 possible configurations of such singularities. We classify
the Hilbert-Burch matrix which corresponds to each configuration. The study of
multiplicity c singularities on, or infinitely near, a fixed rational plane
curve C of degree 2c is equivalent to the study of the scheme of generalized
zeros of the fixed balanced Hilbert-Burch matrix phi for a parameterization of
C.Comment: Typos corrected and minor changes made. To appear in the Memoirs of
the AM
J. Sally's question and a conjecture of Y. Shimoda
In 2007, Y. Shimoda, in connection with a long-standing question of J. Sally,
asked whether a Noetherian local ring, such that all its prime ideals different
from the maximal ideal are complete intersections, has Krull dimension at most
two. In this paper, having reduced the conjecture to the case of dimension
three, if the ring is regular and local of dimension three, we explicitly
describe a family of prime ideals of height two minimally generated by three
elements. Weakening the hypothesis of regularity, we find that, to achieve the
same end, we need to add extra hypotheses, such as completeness, infiniteness
of the residue field and the multiplicity of the ring being at most three. In
the second part of the paper we turn our attention to the category of standard
graded algebras. A geometrical approach via a double use of a Bertini Theorem,
together with a result of A. Simis, B. Ulrich and W.V. Vasconcelos, allows us
to obtain a definitive answer in this setting. Finally, by adapting work of M.
Miller on prime Bourbaki ideals in local rings, we detail some more technical
results concerning the existence in standard graded algebras of homogeneous
prime ideals with an "excessive" number of generators.Comment: 19 pages. Accepted in Nagoya Mathematical Journa
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