Expand all Collapse all | Results 1 - 10 of 10 |
1. CJM 2012 (vol 65 pp. 634)
Laplace Equations and the Weak Lefschetz Property We prove that $r$ independent homogeneous polynomials of the same degree $d$
become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety
whose $(d-1)$-osculating spaces have dimension smaller than expected. This gives an equivalence
between an algebraic notion (called Weak Lefschetz Property)
and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case,
some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture.
Keywords:osculating space, weak Lefschetz property, Laplace equations, toric threefold Categories:13E10, 14M25, 14N05, 14N15, 53A20 |
2. CJM 2012 (vol 65 pp. 961)
A Hilbert Scheme in Computer Vision Multiview geometry is the study of
two-dimensional images of three-dimensional scenes, a foundational subject in computer vision.
We determine a universal GrÃ¶bner basis for the multiview ideal of $n$ generic cameras.
As the cameras move, the multiview varieties vary in a family of dimension $11n-15$.
This family is the distinguished component of a multigraded Hilbert scheme
with a unique Borel-fixed point.
We present a combinatorial study
of ideals lying on that Hilbert scheme.
Keywords:multigraded Hilbert Scheme, computer vision, monomial ideal, Groebner basis, generic initial ideal Categories:14N, 14Q, 68 |
3. CJM 2010 (vol 62 pp. 1246)
Quantum Cohomology of Minuscule Homogeneous Spaces III. Semi-Simplicity and Consequences
We prove that the quantum cohomology ring of any minuscule or
cominuscule homogeneous space, specialized at $q=1$, is semisimple.
This implies that complex conjugation defines an algebra automorphism
of the quantum cohomology ring localized at the quantum
parameter. We check that this involution coincides with the strange
duality defined in our previous article. We deduce Vafa--Intriligator type
formulas for the Gromov--Witten invariants.
Keywords:quantum cohomology, minuscule homogeneous spaces, Schubert calculus, quantum Euler class Categories:14M15, 14N35 |
4. CJM 2008 (vol 60 pp. 961)
About the Defectivity of Certain Segre--Veronese Varieties We study the regularity of the higher secant varieties of $\PP^1\times
\PP^n$, embedded with divisors of type $(d,2)$ and $(d,3)$. We
produce, for the highest defective cases, a ``determinantal'' equation
of the secant variety. As a corollary, we prove that the Veronese
triple embedding of $\PP^n$ is not Grassmann defective.
Keywords:Waring problem, Segre--Veronese embedding, secant variety, Grassmann defectivity Categories:14N15, 14N05, 14M12 |
5. CJM 2008 (vol 60 pp. 875)
A Characterization of the Quantum Cohomology Ring of $G/B$ and Applications We observe that the small quantum product of the
generalized flag manifold $G/B$ is a product operation $\star$ on
$H^*(G/B)\otimes \bR[q_1,\dots, q_l]$ uniquely determined by the
facts
that: it is a deformation of the cup product on $H^*(G/B)$; it is
commutative, associative, and graded with respect to $\deg(q_i)=4$; it
satisfies a certain relation (of degree two); and the corresponding
Dubrovin connection is flat. Previously, we proved that these
properties alone imply the presentation of the ring $(H^*(G/B)\otimes
\bR[q_1,\dots, q_l],\star)$ in terms of generators and relations. In
this paper we use the above observations to give conceptually new
proofs of other fundamental results of the quantum Schubert calculus
for $G/B$: the quantum Chevalley formula of D. Peterson (see also
Fulton and Woodward ) and the ``quantization by standard
monomials" formula of Fomin, Gelfand, and Postnikov for
$G=\SL(n,\bC)$. The main idea of the proofs is the same as in
Amarzaya--Guest: from the quantum $\D$-module of $G/B$ one can
decode all information about the quantum cohomology of this space.
Categories:14M15, 14N35 |
6. CJM 2007 (vol 59 pp. 981)
The Chen--Ruan Cohomology of Weighted Projective Spaces In this paper we study the Chen--Ruan cohomology ring of weighted
projective spaces. Given a weighted projective space ${\bf
P}^{n}_{q_{0}, \dots, q_{n}}$, we determine all of its twisted
sectors and the corresponding degree shifting numbers. The main
result of this paper is that the obstruction bundle over any
3\nobreakdash-multi\-sector is a direct sum of line bundles which we use to
compute the orbifold cup product. Finally we compute the
Chen--Ruan cohomology ring of weighted projective space ${\bf
P}^{5}_{1,2,2,3,3,3}$.
Keywords:Chen--Ruan cohomology, twisted sectors, toric varieties, weighted projective space, localization Categories:14N35, 53D45 |
7. CJM 2007 (vol 59 pp. 488)
Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties We consider the $k$-osculating varieties
$O_{k,n.d}$ to the (Veronese) $d$-uple embeddings of $\PP^n$. We
study the dimension of their higher secant varieties via inverse
systems (apolarity). By associating certain 0-dimensional schemes
$Y\subset \PP^n$ to $O^s_{k,n,d}$ and by studying their Hilbert
functions, we are able, in several cases, to determine whether those
secant varieties are defective or not.
Categories:14N15, 15A69 |
8. CJM 2006 (vol 58 pp. 476)
Apolar Schemes of Algebraic Forms This is a note on the classical Waring's problem for algebraic forms.
Fix integers $(n,d,r,s)$, and let $\Lambda$ be a general $r$-dimensional
subspace of degree $d$ homogeneous polynomials in $n+1$ variables. Let
$\mathcal{A}$ denote the variety of $s$-sided polar polyhedra of $\Lambda$.
We carry out a case-by-case study of the structure of $\mathcal{A}$ for several
specific values of $(n,d,r,s)$. In the first batch of examples, $\mathcal{A}$ is
shown to be a rational variety. In the second batch, $\mathcal{A}$ is a
finite set of which we calculate the cardinality.}
Keywords:Waring's problem, apolarity, polar polyhedron Categories:14N05, 14N15 |
9. CJM 2003 (vol 55 pp. 561)
Quasi-Homogeneous Linear Systems on $\mathbb{P}^2$ with Base Points of Multiplicity $5$ In this paper we consider linear systems of $\mathbb{P}^2$ with all
but one of the base points of multiplicity $5$. We give an explicit
way to evaluate the dimensions of such systems.
Categories:14C20, 14N05 |
10. CJM 1999 (vol 51 pp. 1089)
The Characteristic Numbers of Quartic Plane Curves The characteristic numbers of smooth plane quartics are computed
using intersection theory on a component of the moduli space of
stable maps. This completes the verification of Zeuthen's
prediction of characteristic numbers of smooth plane curves. A
short sketch of a computation of the characteristic numbers of
plane cubics is also given as an illustration.
Categories:14N10, 14D22 |