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Jason Michael Starr

Publications and source records attributed to Jason Michael Starr.

At least 19 recordsLinked to original sources
arXiv2019

Separable rational connectedness and weak approximation in positive characteristic

In this short note we give a characterization of smooth projective varieties of Picard number one that are separably uniruled but not separably rationally connected. We also give a sufficient condition involving the torsion order and the uniruling index for a smooth Fano variety of Picard number one to be separably rationally connected. As an application, we prove some weak approximation results for Fano complete intersections in positive charactersitic. For example, we show that weak approximation holds at place of strong potentially good reduction for a Fano complete intersection in $\mathbb{P}^n$ of type $(d_1, \ldots, d_c)$ in characteristic $p$ such that $n>d_1+\ldots +d_c, p>d_1, \ldots, d_c.$

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arXiv2019

Symplectic invariance of rational surfaces on Kähler manifolds

Kollar and Ruan proved symplectic deformation invariance for uniruledness of Kaehler manifolds. Zhiyu Tian proved the same for rational connectedness in dimension < 4. Kollar conjectured this in all dimensions. We prove Kollar's conjecture, as well as existence of a covering family of rational surfaces, for all Kaehler manifolds that are symplectically deformation equivalent to G/P or to a low degree complete intersection in such.

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arXiv2018

Weak approximation for Fano complete intersections in positive characteristic

For a smooth curve $B$ over an algebraically closed field $k$, for every $B$-flat complete intersection $X_B$ in $B\times_{\text{Spec}\ k} \mathbb{P}^n_k$ of type $(d_1,\dots,d_c)$, if the Fano index is $\geq 2$ and if $\text{char}(k)>\max(d_1,\dots,d_c)$, we prove weak approximation of $\widehat{\mathcal{O}}_{B,b}$-points of $X_B$ by $k(B)$-points at all places of (strong) potentially good reduction, including all places of good reduction. The key step is the proof that such complete intersections are \emph{separably uniruled by lines}, and even \emph{separably rationally connected}, whenever smooth. We prove that the inequality is close to sharp. We prove a similar theorem for Fano manifolds of Picard number $1$ and Fano index $1$.

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arXiv2017

Picard schemes of acyclic schemes

In his work extending rational simple connectedness to schemes with higher Picard rank, Yi Zhu introduced hypotheses for schemes insuring that the relative Picard functor is representable and is étale locally constant with finite free stalks. We give examples showing that one cannot eliminate any of the hypotheses and still have a representable Picard functor that is locally constant with finite free stalks. We also prove that the hypotheses are compatible with composition and with hyperplane sections.

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arXiv2017

Intersection Sheaves for Abel maps

Intersection sheaves, i.e., the Deligne pairing, were first introduced by Deligne in the setting of Poincare duality for etale cohomology, and later in his work on the determinant of cohomology. Intersection sheaves were generalized from smooth schemes to Cohen-Macaulay schemes by Elkik, and then beyond Cohen-Macaulay schemes by Munoz-Garcia. To define the Abel maps arising in rational simple connectedness on the natural parameter spaces of rational sections, we need a variant of the construction of Munoz-Garcia that has the good properties of the construction using det and Div. In addition, we prove basic properties of the classifying stacks that arise in the definition of the Abel maps.

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arXiv2017

Rationally simply connected varieties and pseudo algebraically closed fields

The cohomological dimension of a field is the largest degree with non-vanishing Galois cohomology. Serre's "Conjecture II" predicts that for every perfect field of cohomological dimension $2$, every torsor over the field for a semisimple, simply connected algebraic group is trivial. A field is perfect and "pseudo algebraically closed" (PAC) if every geometrically irreducible curve over the field has a rational point. These have cohomological dimension $1$. Every transcendence degree $1$ extension of such a field has cohomological degree $2$. We prove Serre's "Conjecture II" for such fields of cohomological degree $2$ provided either the field is of characteristic $0$ or the field contains primitive roots of unity for all orders $n$ prime to the characteristic. The method uses "rational simple connectedness" in an essential way. With the same method, we prove that such fields are $C_2$-fields, and we prove that "Period equals Index" for the Brauer groups of such fields. Finally, we use a similar method to reprove and extend a theorem of Fried-Jarden: every perfect PAC field of positive characteristic is $C_2$

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arXiv2017

Veronese varieties contained in hypersurfaces

Alex Waldron proved that for sufficiently general degree $d$ hypersurfaces in projective $n$-space, the Fano scheme parameterizing $r$-dimensional linear spaces contained in the hypersurface is nonempty precisely for the degree range $n\geq N_1(r,d)$ where the "expected dimension" $f_1(n,r,d)$ is nonnegative, in which case $f_1(n,r,d)$ equals the (pure) dimension. Using work by Gleb Nenashev, we prove that for sufficiently general degree $d$ hypersurfaces in projective $n$-space, the parameter space of $r$-dimensional $e$-uple Veronese varieties contained in the hypersurface is nonempty of pure dimension equal to the "expected dimension" $f_e(n,r,d)$ in a degree range $n\geq \widetilde{N}_e(r,d)$ that is asymptotically sharp. Moreover, we show that for $n\geq 1+N_1(r,d)$, the Fano scheme parameterizing $r$-dimensional linear spaces is irreducible.

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arXiv2008

Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

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 algebro-geometric 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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arXiv2006

Fano varieties and linear sections of hypersurfaces

Under a hypothesis on $k$, $d$ and $n$ that is almost the best possible, we prove that for every smooth degree $d$ hypersurface in $P^n$, the $k$-plane sections dominate the moduli space of degree $d$ hypersurface in $P^k$. Using this we prove rational simple connectedness of every smooth degree $d$ hypersurface in $P^n$, under a suitable hypothesis on $d$ and $n$ (previous results were only for general hypersurfaces).

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arXiv2006

A pencil of Enriques surfaces of index one with no section

Mondromy and deformation-and-specialization arguments are used to produce a pencil of Enriques surfaces with no section and index one. The same technique completes the proof that the witness curves in "Rational connectivity ..." depend on the relative dimension.

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arXiv2006

Very twisting families of pointed lines on Grassmannians

There exist very twisting families of lines on Grassmannians and isotropic Grassmannians. This is a key step in proving existence of a rational section of a family of isotropic Grassmannians over a surface with vanishing Brauer obstruction.

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arXiv2006

Artin's axioms, composition and moduli spaces

We prove Artin's axioms satisfy a compatibility for composition of 1-morphisms of stacks in groupoids. Consequently, some natural stacks in groupoids are algebraic, including a common generalization of Vistoli's Hilbert stack and the stack of branchvarieties of Alexeev and Knutson.

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arXiv2006

Higher Fano manifolds and rational surfaces

For a Fano manifold of pseudo-index at least 3 and $c_1^2-2c_2$ nef, we show irreducibility of certain spaces of curves on the Fano manifold implies the manifold is a union of rational surfaces.

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