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Journals and Conferences
We establish a positive geometric rule for computing the structure constants of the cohomology of two-step flag varieties with respect to their Schubert basis. As a corollary we obtain a quantum LittlewoodRichardson rule for Grassmannians. These rules have numerous applications to geometry, representation theory and the theory of symmetric functions.
In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps M0,0(P , d) stabilize when r ≥ d. We give a complete characterization of the effective divisors on M0,0(P, d): They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.
We prove that the space of rational curves of a fixed degree on any smooth cubic hypersurface of dimension at least four is irreducible and of the expected dimension. Our methods also show that the space of rational curves of a fixed degree on a general hypersurface in Pn of degree 2d ≤ min(n+4, 2n−2) and dimension at least three is irreducible and of the… (More)
We describe recent work on positive descriptions of the structure constants of the cohomology of homogeneous spaces such as the Grassmannian, by degenerations and related methods. We give various extensions of these rules, some new and conjectural, to K-theory, equivariant cohomology, equivariant K-theory, and quantum cohomology.
We describe an algorithm for computing certain characteristic numbers of rational normal surface scrolls using degenerations. As a corollary we obtain an efficient method for computing the corresponding Gromov-Witten invariants of the Grassmannians of lines.
A Schubert class in the Grassmannian is rigid if the only proper subvarieties representing that class are Schubert varieties. The hyperplane class σ1 is not rigid because a codimension one Schubert cycle can be deformed to a smooth hyperplane section. In this paper, we show that this phenomenon accounts for the failure of rigidity in Schubert classes. More… (More)
This paper develops a new method for studying the cohomology of orthogonal flag varieties. Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to (not necessarily isotropic) flags. They interpolate between Schubert varieties in orthogonal flag varieties and the restrictions of general Schubert… (More)
Given a zero-dimensional scheme Z, the higher-rank interpolation problem asks for the classification of slopes μ such that there exists a vector bundle E of slope μ satisfying H(E ⊗ IZ) = 0 for all i. In this paper, we solve this problem for all zero-dimensional monomial schemes in P. As a corollary, we obtain detailed information on the stable base loci of… (More)
We produce ample, respectively NEF, eventually free, divisors in the Kontsevich space M0,n(P , d) of n-pointed, genus 0, stable maps to Pr, given such divisors in M0,n+d. We prove this produces all ample, respectively NEF, eventually free, divisors in M0,n(P, d). As a consequence, we construct a contraction of the boundary ∪ bd/2c k=1 ∆k,d−k in M0,0(P r, d)… (More)
In this paper, we determine the stable base locus decomposition of the Kontsevich moduli spaces of degree two and three stable maps to Grassmannians. This gives new examples of the decomposition for varieties with Picard rank three. We also discuss the birational models that correspond to the chambers in the decomposition.