The purpose of this paper is to compute the minimal fibering degree of an arbitrary projective toric variety. We prove that it equals the lattice width of the associated polytope. This gives a complete answer to a question asked in a recent paper of Levinson, Ullery and the second author. The minimal fibering degree of a polarized projective variety was introduced in that paper in order to compute the degree of irrationality (a generalization of gonality) of high degree divisors. From this perspective, our paper gives a higher dimensional analogue of results of Kawaguchi and others who computed the gonality of curves in toric surfaces in terms of lattice widths.

For prime degree hypersurfaces of dimension at least 3, Mori asked if every smooth proper limit is still a hypersurface. Interestingly in dimensions 1 and 2, this is not the case. For example, Griffin constructed explicit families of quintic curves that give counterexamples in dimension 1 (Horikawa constructed similar examples for quintics in dimension 2). In this paper we propose a conjecture explaining these examples using Hacking and Prokhorov's work on Q-Gorenstein limits of the projective plane. In particular, if p is a prime number that is not a Markov number, we conjecture that any smooth projective limit of plane curves of degree p is a plane curve. The main results are to prove this conjecture for degree 7 curves and to extend Griffin's counterexamples to all prime numbers that are also Markov numbers.

We show that the degrees of rational endomorphisms of very general complex Fano and Calabi-Yau hypersurfaces satisfy certain congruence conditions by specializing to characteristic p. As a corollary we show that very general n-dimensional hypersurfaces of degree $d\ge 5\lceil (n+3)/6\rceil$ are not birational to elliptic fibrations. A key part of the argument is to resolve singularities of general p-cyclic covers in mixed characteristic p.

Koll\'ar proved that a very general $n$-dimensional complex hypersurface of degree at least $3\lceil (n+3)/4\rceil$ is not birational to a fibration in rational curves. This is most interesting when the hypersurface is Fano, in which case it is covered by rational curves. In this paper, we extend Koll\'ar's ideas and show that for any genus $g$, there are Fano hypersurfaces (in more restrictive degree and dimension ranges) that are not birational to fibrations in genus $g$ curves. In other words, we show that the fibering genus of these hypersurfaces can be arbitrarily large. The fibering genus of a variety has been studied in work of Konno, Ein--Lazarsfeld, and Voisin, but this is the first paper to explore these ideas in the Fano range. Following Koll\'ar, we degenerate to characteristic $p>0$ to rule out these fibrations. A crucial input is Tate's genus change formula and its generalizations, which imply that any regular curve of genus $g$ is smooth if $p$ is sufficiently large compared to $g$.

In this paper we study the degrees of irrationality of hypersurfaces of large degree in a complex projective variety. We show that the maps computing the degrees of irrationality of these hypersurfaces factor through rational fibrations of the ambient variety. As a consequence, we give tight bounds on the degree of irrationality of these hypersurfaces in terms of a new invariant of independent interest: the minimal fibering degree of a projective variety with respect to an effective divisor. As a corollary we show that the degree of irrationality of a complete intersection of sufficiently large and unbalanced degrees is roughly the product of the degrees. This gives a partial answer to a question of Bastianelli, De Poi, Ein, Lazarsfeld, and the third author.

We use the specialization homomorphism for the birational automorphism group to study finite order birational automorphisms. For a family of varieties over a DVR, we prove that a birational automorphism of order coprime to the residue characteristic cannot specialize to the identity. As an application, we show that very general $n$-dimensional hypersurfaces of degree $d \geq 5 \lceil (n+3)/6 \rceil$ have no finite order birational automorphisms.

A famous problem in birational geometry is to determine when the birational automorphism group of a Fano variety is finite. The Noether–Fano method has been the main approach to this problem. The purpose of this paper is to give a new approach to the problem by showing that in every positive characteristic, there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphism group. To do this, we prove that these varieties admit ample and birationally equivariant line bundles. Our result applies the differential forms that Kollár produces on $p$-cyclic covers in characteristic $p > 0$.

Motivated by the study of rationally connected fibrations, we study different notions of birationally simple fibrations. Our main result is the construction of maximal Chow constant and cohomologically constant fibrations. This paper is largely self-contained and we prove a number of basic properties of these fibrations. One application is to the classification of "rationalizations of singularities of cones." We also consider consequences for the Chow groups of the generic fiber of a Chow constant fibration.

Hartshorne's conjecture about vector bundles on projective space states that any rank 2 vector bundle on n-dimensional projective space splits as soon as n is at least 7. Klyachko has shown that Hartshorne's conjecture is true when the vector bundles are torus equivariant. Moreover, recent work of Ilten and S\"uss generalizes Klyachko's work to the case of a smaller rank torus action on projective space. In this note we give a new, direct proof that torus rank 2 bundles split that avoids a description of the category of torus equivariant vector bundles.

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index $e$, then the degree of irrationality of a very general complex Fano hypersurface of index $e$and dimension n is bounded from below by a constant times $\sqrt{n}$. To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic$p$argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.

The point of this paper is to give a short, direct proof that rank $2$ toric vector bundles on $n$-dimensional projective space split once $n$ is at least $3$. This result is originally due to Bertin and Elencwajg, and there is also related work by Kaneyama, Klyachko, and Ilten-Süss. The idea is that, after possibly twisting the vector bundle, there is a section which is a complete intersection.

The purpose of this paper is to compute the degree of irrationality of hypersurfaces of sufficiently high degree in various Fano varieties: quadrics, Grassmannians, products of projective space, cubic threefolds, cubic fourfolds, and complete intersection threefolds of type (2,2). This extends the techniques of Bastianelli, De Poi, Ein, Lazarsfeld, and the second author who computed the degree of irrationality of hypersurfaces of sufficiently high degree in projective space. A theme in the paper is that the fibers of low degree rational maps from the hypersurfaces to projective space tend to lie on curves of low degree contained in the Fano varieties. This allows us to study these maps by studying the geometry of curves in these Fano varieties.

The purpose of this paper is to study the Zariski tangent space of the punctual Hilbert scheme parametrizing subschemes of a smooth surface which are supported at a single point. We give a lower bound on the dimension of the tangent space in general and show the bound is sharp for subschemes of the affine plane cut out by monomials. Furthermore for monomial subschemes we give an explicit combinatorial formula for the dimension of the tangent space.

of the Dissertation The Degree of Irrationality of Very General Hypersurfaces in Some Homogeneous Spaces

The purpose of this paper is to explore the geometry and establish the slope stability of tautological vector bundles on Hilbert schemes of points on smooth surfaces. By establishing stability in general we complete a series of results of Schlickewei and Wandel who proved the slope stability of these vector bundles for Hilbert schemes of 2 points or 3 points on K3 or abelian surfaces with Picard group restrictions. In exploring the geometry we show that every sufficiently positive semistable vector bundle on a smooth curve arises as the restriction of a tautological vector bundle on the Hilbert scheme of points on the projective plane. Moreover we show the tautological bundle of the tangent bundle is naturally isomorphic to the sheaf of vector fields tangent to the divisor which consists of nonreduced subschemes.