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We prove that the Chow quotient parameterizing configurations of n n points in P d \mathbb {P}^d which generically lie on a rational normal curve is isomorphic to M ¯ 0 , n \overline {\mathcal {M}}_{0,n} , generalizing the well-known d = 1 d=1 result of Kapranov. In particular, M ¯ 0 , n \overline {\mathcal {M}}_{0,n} admits birational morphisms to all the corresponding geometric invariant theory (GIT) quotients. For symmetric linearizations, the polarization on each GIT quotient pulls back to a divisor that spans the same extremal ray in the symmetric nef cone of M ¯ 0 , n \overline {\mathcal {M}}_{0,n} as a conformal blocks line bundle. A symmetry in conformal blocks implies a duality of point-configurations that comes from Gale duality and generalizes a result of Goppa in algebraic coding theory. In a suitable sense, M ¯ 0 , 2 m \overline {\mathcal {M}}_{0,2m} is fixed pointwise by the Gale transform when d = m − 1 d=m-1 so stable curves correspond to self-associated configurations.

In [Math. Ann. 142, 453-468], Remmert and Van de Ven conjectured that if X X is the image of a surjective holomorphic map from P n \mathbb {P}^n , then X X is biholomorphic to P n \mathbb {P}^n . This conjecture was proved by Lazarsfeld [Lect. Notes Math. 1092 (1984), 29-61] using Mori’s proof of Hartshorne’s conjecture [Ann. Math. 110 (1979), 593-606]. Then Lazarsfeld raised a more general problem, which was completely answered in the positive by Hwang and Mok. Theorem 1 ([Invent. math. 136 (1999), 209-231] and [Asian J. Math. 8 (2004), 51-63]). Let S = G / P S=G/P be a rational homogeneous manifold of Picard number 1 1 . For any surjective holomorphic map f : S → X f:S\to X to a projective manifold X X , either X X is a projective space, or f f is a biholomorphism. The aim of this article is to give a generalization of Theorem 1. We will show that modulo canonical projections, Theorem 1 is true when G G is simple without the assumption on Picard number. We need to find a dominating and generically unsplit family of rational curves which are of positive degree with respect to a given nef line bundle on X X . Such a family may not exist in general, but we will prove its existence under a certain assumption which is applicable in our situation.

The aim of this paper is to present a construction of smooth rational surfaces in projective fourspace with degree 12 12 and sectional genus 13 13 . In particular, we establish the existences of five different families of smooth rational surfaces in projective fourspace with the prescribed invariants.

The uniform position principle states that, given an irreducible nondegenerate curve C in the projective r-space $P^r$, a general (r-2)-plane L is uniform, that is, projection from L induces a rational map from C to $P^1$ whose monodromy group is the full symmetric group. In this paper we show the locus of non-uniform (r-2)-planes has codimension at least two in the Grassmannian for a curve C with arbitrary singularities. This result is optimal in $P^2$. For a smooth curve C in $P^3$ that is not a rational curve of degree three, four or six, we show any irreducible surface of non-uniform lines is a Schubert cycle of lines through a point $x$, such that projection from $x$ is not a birational map of $C$ onto its image., corrected typo in first paragraph of introduction, 23 pages, AMSLaTeX

Let X be a projective variety which is covered by a family of rational curves of minimal degree. The classic bend-and-break argument of Mori asserts that if x and y are two general points, then there are at most finitely many curves in that family which contain both x and y. In this work we shed some light on the question as to whether two sufficiently general points actually define a unique curve. As an immediate corollary to the results of this paper, we give a characterization of projective spaces which improves on the known generalizations of Kobayashi-Ochiai's theorem., Comment: Reason for resubmission: improved exposition