1,677 research outputs found
The geometry of eight points in projective space: Representation theory, Lie theory, dualities
This paper deals with the geometry of the space (GIT quotient) M_8 of 8
points in P^1, and the Gale-quotient N'_8 of the GIT quotient of 8 points in
P^3. The space M_8 comes with a natural embedding in P^{13}, or more precisely,
the projectivization of the S_8-representation V_{4,4}. There is a single
S_8-skew cubic C in P^{13}. The fact that M_8 lies on the skew cubic C is a
consequence of Thomae's formula for hyperelliptic curves, but more is true: M_8
is the singular locus of C. These constructions yield the free resolution of
M_8, and are used in the determination of the "single" equation cutting out the
GIT quotient of n points in P^1 in general. The space N'_8 comes with a natural
embedding in P^{13}, or more precisely, PV_{2,2,2,2}. There is a single skew
quintic Q containing N'_8, and N'_8 is the singular locus of the skew quintic
Q. The skew cubic C and skew quintic Q are projectively dual. (In particular,
they are surprisingly singular, in the sense of having a dual of remarkably low
degree.) The divisor on the skew cubic blown down by the dual map is the secant
variety Sec(M_8), and the contraction Sec(M_8) - - > N'_8 factors through N_8
via the space of 8 points on a quadric surface. We conjecture that the divisor
on the skew quintic blown down by the dual map is the quadrisecant variety of
N'_8 (the closure of the union of quadrisecant *lines*), and that the quintic Q
is the trisecant variety. The resulting picture extends the classical duality
in the 6-point case between the Segre cubic threefold and the Igusa quartic
threefold. We note that there are a number of geometrically natural varieties
that are (related to) the singular loci of remarkably singular cubic
hypersurfaces. Some of the content of this paper appeared in arXiv/0809.1233.Comment: 31 pages, 4 figure
Schur Q-functions and degeneracy locus formulas for morphisms with symmetries
We give closed-form formulas for the fundamental classes of degeneracy loci
associated with vector bundle maps given locally by (not necessary square)
matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal.
Our description uses essentially Schur Q-polynomials of a bundle, and is based
on a certain push-forward formula for these polynomials in a Grassmann bundle.Comment: 22 pages, AMSTEX, misprints corrected, exposition improved. to appear
in the Proceedings of Intersection Theory Conference in Bologna, "Progress in
Mathematics", Birkhause
Orbitopes
An orbitope is the convex hull of an orbit of a compact group acting linearly
on a vector space. These highly symmetric convex bodies lie at the crossroads
of several fields, in particular convex geometry, optimization, and algebraic
geometry. We present a self-contained theory of orbitopes, with particular
emphasis on instances arising from the groups SO(n) and O(n). These include
Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese
orbitopes and Grassmann orbitopes. We study their face lattices, their
algebraic boundary hypersurfaces, and representations as spectrahedra or
projected spectrahedra.Comment: 37 pages. minor revisions of origina
Braided affine geometry and q-analogs of wave operators
The main goal of this review is to compare different approaches to
constructing geometry associated with a Hecke type braiding (in particular,
with that related to the quantum group U_q(sl(n))). We make an emphasis on
affine braided geometry related to the so-called Reflection Equation Algebra
(REA). All objects of such type geometry are defined in the spirit of affine
algebraic geometry via polynomial relations on generators.
We begin with comparing the Poisson counterparts of "quantum varieties" and
describe different approaches to their quantization. Also, we exhibit two
approaches to introducing q-analogs of vector bundles and defining the
Chern-Connes index for them on quantum spheres. In accordance with the
Serre-Swan approach, the q-vector bundles are treated as finitely generated
projective modules over the corresponding quantum algebras.
Besides, we describe the basic properties of the REA used in this
construction and compare different ways of defining q-analogs of partial
derivatives and differentials on the REA and algebras close to them. In
particular, we present a way of introducing a q-differential calculus via
Koszul type complexes. The lements of the q-calculus are applied to defining
q-analogs of some relativistic wave operators.Comment: A review submitted to Journal of Physics A: Mathematical and
Theoretica
- …