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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 P n of degree 2d ≤ min(n + 4, 2n − 2) and dimension at least three is irreducible and of… (More)
We describe recent work on positive descriptions of the structure constants of the coho-mology 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.
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.
This paper investigates low-codimension degenerations of Del Pezzo surfaces. As an application we determine certain characteristic numbers of Del Pezzo surfaces. Finally, we analyze the relation between the enumerative geometry of Del Pezzo surfaces and the Gromov-Witten invariants of the Hilbert scheme of conics in P N .
We introduce and compute the class of a number of effective divisors on the moduli space of stable maps M 0,0 (P r ,d), which, for small d, provide a good understanding of the extremal rays and the stable base locus decomposition for the effective cone. We also discuss various birational models that arise in Mori's program, including the Hilbert scheme, the… (More)
Given a vector bundle E on a smooth projective variety X, we can define subschemes of the Kontsevich moduli space of genus-zero stable maps M 0,0 (X, β) parameterizing maps f : P 1 → X such that the Grothendieck decomposition of f * E has a specified splitting type. In this paper, using a " compactification " of this locus, we define Gromov-Witten… (More)
Curves of genus greater than two exhibit arithmetic and geometric properties very different from curves of genus zero and one. In these notes we will survey a few ways of extrapolating these differences to higher dimensional complex man-ifolds. We will specifically concentrate on Brody and Kobayashi hyperbolicity, which are generalizations of complex… (More)