Jason Starr

Families of rationally connected varieties

Recall that a proper variety X is said to be rationally connected if two general points p, q ∈ X are contained in the image of a map g : P → X. This is clearly a birationally invariant property. When X is smooth, this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset Γ ⊂ X, there is a map g : P → X whose image contains Γ and such that gTX is an ample bundle.

Quot Functors for Deligne-Mumford Stacks

Abstract Given a separated and locally finitely-presented Deligne-Mumford stack 𝒳 over an algebraic space S, and a locally finitely-presented 𝒪𝒳-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. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Higher Fano manifolds and rational surfaces

Let X be a Fano manifold of pseudo-index ≥ 3 such that c1(X)− 2c2(X) is nef. Irreducibility of some spaces of rational curves on X implies a general point of X is contained in a rational surface.

Divisors on the space of maps to Grassmannians

In this note we study the divisor theory of the Kontsevich moduli spacesM0,0(G(k, n), d) of genus-zero stable maps to the Grassmannians. We calculate the classes of several geometrically significant divisors. We prove that the cone of effective divisors stabilizes as n increases and we determine the stable effective cone. We also characterize the ample cone.

Rational surfaces in index-one Fano hypersurfaces

We give the first evidence for a conjecture that a general, index-one, Fano hypersurface is not unirational: (i) a general point of the hypersurface is contained in no rational surface ruled, roughly, by low-degree rational curves, and (ii) a general point is contained in no image of a Del Pezzo surface.

Arithmetic Questions Related to Rationally Connected Varieties

Let X ⊂ ℙ n be a nonempty variety defined over a field K. Under what combination of conditions on the geometry of X and the algebra of K can we be sure that X has a K-rational point?

Divisor Classes and the Virtual Canonical Bundle for Genus 0 Maps

We prove divisor class relations for families of genus 0 curves and used them to compute the divisor class of the “virtual” canonical bundle of the Kontsevich space of genus 0 maps to a smooth target. This agrees with the canonical bundle in good cases. This work generalizes Pandharipande’s results in the special case that the target is projective space, [7] (Pandharipande, Trans. Am. Math. Soc. 351(4), 1481–1505, 1999), [8] (Pandharipande, Trans. Am. Math. Soc. 351(4), 1481–1505, 1999). Our method is completely different from Pandharipande’s.

Jumps in Mordell-Weil Rank and Arithmetic Surjectivity

We ask the question: If a pencil of curves of genus one defined over Q admits no section, can we find a number field L/Q and a member of that pencil defined over L having no L-rational points?

SEMISTABLE LOCUS OF A GROUP COMPACTIFICATION

In this paper, we consider the diagonal action of a connected semisimple group of adjoint type on its wonderful compactification. We show that the semi-stable locus is a union of the

Every rationally connected variety over the function field of a curve has a rational point

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