We prove that every one-parameter family of complex rationally connected varieties has a section.

Abstract. Given a separated and locally finitely-presented DeligneMumford stack X over an algebraic space S, and a locally finitelypresented O X -module F , we prove that the Quot functor Quot(F /X /S) is represented by a separated and locally finitely-presented algebraic space over S. Under additional hypotheses, we prove that the connected components of Quot(F /X /S) are quasi-projective over S.

Abstract. We prove that for n > 2 and d < n+1 2 , a general complex hypersurface X ⊂ P n of degree d has the property that for each integer e the scheme R e (X) parametrizing degree e, smooth rational curves on X is an integral, local complete intersection scheme of "expected" dimension (n + 1 − d)e + (n − 4).The techniques used in the proof include: 1. Classical results about lines on hypersurfaces including a new result about flatness of the projection map from the space of pointed lines. 2. The Kontsevich moduli space of stable maps, M 0,r (X, e). In particular we use the deformation theory of stable maps, properness of the stack M 0,r (X, e), and the decomposition of M 0,r (X, e) described in [2]. 3. A version of Mori's bend-and-break lemma.

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Abstract. Under suitable hypotheses, we prove that a form of a projective homogeneous variety G/P defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebrogeometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre's Conjecture II in Galois cohomology for function fields over an algebraically closed field.

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In a paper from 1992, Kollár, Miyaoka and Mori posed the following question: Given a proper flat morphism f : P → X with target a nonsingular curve and whose geometric generic fiber is rationally-connected, does it follow that f admits a regular section? In the case that the ground field is an algebraically-closed field of characteristic zero, this was answered affirmatively by Graber, Harris, and Starr using a topological argument. We prove that f admits a regular section when the ground field is an algebraically-closed field of arbitrary characteristic. The key ingredients in the proof are the following: (a) Proposition 2.1 which shows that after taking the reduced fiber product of f with a generically étale morphism π: Y → X , we may assume that the smooth locus of f intersects every geometric fiber, and (b) Proposition 1.1 which is a purely algebraic analogue of the "moving branch points" argument in Graber-Harris-Starr. The reader is cautioned that our definition of "separably rationally connected" differs slightly from the original definition.

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