Expand all Collapse all | Results 1 - 4 of 4 |
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 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 |
3. 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 |
4. 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 |