1,157 research outputs found
Koszul algebras and regularity
This is a survey paper on commutative Koszul algebras and Castelnuovo-Mumford
regularity. We describe several techniques to establish the Koszulness of
algebras. We discuss variants of the Koszul property such as strongly Koszul,
absolutely Koszul and universally Koszul. We present several open problems
related with these notions and their local variants
Common transversals and tangents to two lines and two quadrics in P^3
We solve the following geometric problem, which arises in several
three-dimensional applications in computational geometry: For which
arrangements of two lines and two spheres in R^3 are there infinitely many
lines simultaneously transversal to the two lines and tangent to the two
spheres?
We also treat a generalization of this problem to projective quadrics:
Replacing the spheres in R^3 by quadrics in projective space P^3, and fixing
the lines and one general quadric, we give the following complete geometric
description of the set of (second) quadrics for which the 2 lines and 2
quadrics have infinitely many transversals and tangents: In the
nine-dimensional projective space P^9 of quadrics, this is a curve of degree 24
consisting of 12 plane conics, a remarkably reducible variety.Comment: 26 pages, 9 .eps figures, web page with more pictures and and archive
of computations: http://www.math.umass.edu/~sottile/pages/2l2s
Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves
We give examples over arbitrary fields of rings of invariants that are not
finitely generated. The group involved can be as small as three copies of the
additive group, as in Mukai's examples over the complex numbers. The failure of
finite generation comes from certain elliptic fibrations or abelian surface
fibrations having positive Mordell-Weil rank.
Our work suggests a generalization of the Morrison-Kawamata cone conjecture
from Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture in
dimension 2 in the case of minimal rational elliptic surfaces.Comment: 26 pages. To appear in Compositio Mathematic
Signature Sequence of Intersection Curve of Two Quadrics for Exact Morphological Classification
We present an efficient method for classifying the morphology of the
intersection curve of two quadrics (QSIC) in PR3, 3D real projective space;
here, the term morphology is used in a broad sense to mean the shape,
topological, and algebraic properties of a QSIC, including singularity,
reducibility, the number of connected components, and the degree of each
irreducible component, etc. There are in total 35 different QSIC morphologies
with non-degenerate quadric pencils. For each of these 35 QSIC morphologies,
through a detailed study of the eigenvalue curve and the index function jump we
establish a characterizing algebraic condition expressed in terms of the Segre
characteristics and the signature sequence of a quadric pencil. We show how to
compute a signature sequence with rational arithmetic so as to determine the
morphology of the intersection curve of any two given quadrics. Two immediate
applications of our results are the robust topological classification of QSIC
in computing B-rep surface representation in solid modeling and the derivation
of algebraic conditions for collision detection of quadric primitives
Real k-flats tangent to quadrics in R^n
Let d_{k,n} and #_{k,n} denote the dimension and the degree of the
Grassmannian G_{k,n} of k-planes in projective n-space, respectively. For each
k between 1 and n-2 there are 2^{d_{k,n}} \cdot #_{k,n} (a priori complex)
k-planes in P^n tangent to d_{k,n} general quadratic hypersurfaces in P^n. We
show that this class of enumerative problem is fully real, i.e., for each k
between 1 and n-2 there exists a configuration of d_{k,n} real quadrics in
(affine) real space R^n so that all the mutually tangent k-flats are real.Comment: 10 pages, 3 figures. Minor revisions, to appear in Proc. AM
Computing the Chow variety of quadratic space curves
Quadrics in the Grassmannian of lines in 3-space form a 19-dimensional
projective space. We study the subvariety of coisotropic hypersurfaces.
Following Gel'fand, Kapranov and Zelevinsky, it decomposes into Chow forms of
plane conics, Chow forms of pairs of lines, and Hurwitz forms of quadric
surfaces. We compute the ideals of these loci
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