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Search: MSC category 14N15 ( Classical problems, Schubert calculus )

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1. CJM Online first

Levinson, Jake
One-dimensional Schubert problems with respect to osculating flags
We consider Schubert problems with respect to flags osculating the rational normal curve. These problems are of special interest when the osculation points are all real -- in this case, for zero-dimensional Schubert problems, the solutions are "as real as possible". Recent work by Speyer has extended the theory to the moduli space $ \overline{\mathcal{M}_{0,r}} $, allowing the points to collide. These give rise to smooth covers of $ \overline{\mathcal{M}_{0,r}} (\mathbb{R}) $, with structure and monodromy described by Young tableaux and jeu de taquin. In this paper, we give analogous results on one-dimensional Schubert problems over $ \overline{\mathcal{M}_{0,r}} $. Their (real) geometry turns out to be described by orbits of Schützenberger promotion and a related operation involving tableau evacuation. Over $\mathcal{M}_{0,r}$, our results show that the real points of the solution curves are smooth. We also find a new identity involving "first-order" K-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof.

Keywords:Schubert calculus, stable curves, Shapiro-Shapiro Conjecture, jeu de taquin, growth diagram, promotion
Categories:14N15, 05E99

2. CJM 2012 (vol 65 pp. 634)

Mezzetti, Emilia; Miró-Roig, Rosa M.; Ottaviani, Giorgio
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

3. CJM 2008 (vol 60 pp. 961)

Abrescia, Silvia
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

4. CJM 2007 (vol 59 pp. 488)

Bernardi, A.; Catalisano, M. V.; Gimigliano, A.; Idà, M.
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

5. CJM 2006 (vol 58 pp. 476)

Chipalkatti, Jaydeep
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

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