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Search: MSC category 14N ( Projective and enumerative geometry [See also 51-XX] )

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1. CJM 2016 (vol 69 pp. 143)

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 2012 (vol 65 pp. 961)

Aholt, Chris; Sturmfels, Bernd; Thomas, Rekha
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 Gröbner 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

4. CJM 2010 (vol 62 pp. 1246)

Chaput, P. E.; Manivel, L.; Perrin, N.
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

5. 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

6. CJM 2008 (vol 60 pp. 875)

Mare, Augustin-Liviu
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

7. CJM 2007 (vol 59 pp. 981)

Jiang, Yunfeng
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

8. 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

9. 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

10. CJM 2003 (vol 55 pp. 561)

Laface, Antonio; Ugaglia, Luca
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

11. CJM 1999 (vol 51 pp. 1089)

Vakil, Ravi
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

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