arXiv subjects
Publications and source records attributed to Nathan Chen.
The purpose of this paper is to study low degree points on plane curves. We prove results analogous to those of Debarre and Klassen for singular plane curves with a finite number $\delta$ of ordinary nodes/cusps, where $\delta$ is bounded from above by a quadratic function in the degree of the plane curve.
Suppose $X$ is an irreducible complex variety. We show that when $X$ is ruled, the group of birational transformations $Bir(X)$, as a group, determines $X$ up to birational transformations and automorphisms of the base field. In contrast, we demonstrate that this same property never holds for non-uniruled varieties.
Zariski dense collections of quadratic points on curves $X$ are well-understood by results of Harris--Silverman and Vojta, but when $\dim X \geq 2$ there is not an analogous geometric characterization, even conjecturally. In this note we consider the case of a double cover $\pi \colon X \to \mathbb{P}^r$, where Hilbert's Irreducibility Theorem implies that the quadratic points in the fibers of $\pi$ are dense. We show that Vojta's Conjecture implies that, once the canonical bundle of $X$ is sufficiently positive, there are no other sources of Zariski dense quadratic points. This is complemented by several examples of surfaces $X \to \mathbb{P}^2$ with an additional source of dense quadratic points.
The purpose of this note is to show that the subvarieties of small degree inside a general hypersurface of large degree come from intersecting with linear spaces or other varieties.
Measures of irrationality are a numerical way of quantifying how far a given variety is from being rational (or rationally connected, uniruled, etc.). In the last two decades, there has been renewed interest in the study of these invariants. The goal of this expository survey is to summarize known results from the point of view of the Kodaira--Enriques classification of surfaces, highlight recent progress, and discuss a number of open problems and questions.
Rationality specializes in families of surfaces, even with mild singularities. In this paper, we study the analogous question for the degree of irrationality. We prove a specialization result when the degree of irrationality on the generic fiber arises from the quotient by a group action.
We study the minimal degrees and gonalities of curves on complete intersections. We prove that the degree of any curve on a general complete intersection $X \subseteq \mathbb{P}^N$ of large multidegree is bounded from below by the degree of $X$. As an application, we answer a problem of Bastianelli--De Poi--Ein--Lazarsfeld--Ullery on measures of irrationality for complete intersections.
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 $g$.
A result of Popa and Schnell shows that any holomorphic 1-form on a smooth complex projective variety of general type admits zeros. More generally, given a variety $X$ which admits $g$ pointwise linearly independent holomorphic 1-forms, their result shows that $X$ has Kodaira dimension $\kappa(X) \leq \dim X - g$. In the extremal case where $\kappa(X) = \dim X - g$ and $X$ is minimal, we prove that $X$ admits a smooth morphism to an abelian variety, and classify all such $X$ by showing they arise as diagonal quotients of the product of an abelian variety with a variety of general type. The case $g = 1$ was first proved by the third author, and classification results about surfaces and threefolds carrying nowhere vanishing forms have appeared in work of Schreieder and subsequent joint work with the third author. We also prove a birational version of this classification which holds without the minimal assumption, and establish additional cases of a conjecture of 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.
We study dominant rational maps from a product of two curves to surfaces with $p_{g} = q = 0$. Given two curves which satisfy a mild genericity assumption and have large genus relative to their gonality, we show that the degree of irrationality of their product is equal to the product of their gonalities. Moreover, we prove that the degree of irrationality of a product of two hyperelliptic curves is 4.
We show that measures of irrationality on very general codimension two complete intersections and very general complete intersection surfaces are multiplicative in the degrees of the defining equations. This confirms some cases of a conjecture of Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery. Our methods involve studying the numerical invariants of curves on complete intersections.
Determining when the birational automorphism group of a Fano variety is finite is an interesting and difficult problem. The main technique for studying this problem is by the Noether-Fano method. This method has been effective in studying this problem for Fano varieties of index one and two. The purpose of this paper is to give a new approach to this problem, and to show that in every positive characteristic there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphisms. To do this we prove that these varieties admit ample and birationally equivariant line bundles. Our result applies the differential forms that Koll\'ar produces on p-cyclic covers in characteristic p>0.
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.
Let $(X,L)$ be a general primitively polarized K3 surface with $c_1(L)^2 = 2g-2$ for some integer $g \geq 2$. The Severi variety $V^{L,\delta} \subset |L|$ is defined to be the locus of reduced and irreducible curves in $|L|$ with exactly $\delta$ nodes and no other singularities. When $\delta=g$, any curve $C \in V^{L,g}$ is a rational curve; in fact, Chen \cite{Chen02} has shown that all rational curves in $|L|$ are nodal, and the number of such rational curves is given by the Yau-Zaslow formula \cite{YZ96}. In this paper, we consider the next case where $\delta = g-1$ and the Severi variety $V^{L,g-1}$ parametrizing nodal elliptic curves is of dimension 1. Let $\overline{V}^{L,g-1} \subset |L|$ denote the Zariski closure. For a reduced curve $C$, we define the geometric genus of $C$ to be the sum of the genera of the irreducible components of the normalization. We prove that the geometric genus of the closure $\overline{V}^{L,g-1} \subset |L|$ is bounded from below by $O(e^{C\sqrt{g}})$.
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 Koll\'ar 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 degree of irrationality of a projective variety $X$ is defined to be the smallest degree rational dominant map to a projective space of the same dimension. For abelian surfaces, Yoshihara computed this invariant in specific cases, while Stapleton gave a sublinear upper bound for very general polarized abelian surfaces $(A, L)$ of degree $d$. Somewhat surprisingly, we show that the degree of irrationality of a very general polarized abelian surface is uniformly bounded above by $4$, independently of the degree of the polarization. This result disproves part of a conjecture of Bastianelli, De Poi, Ein, Lazarsfeld and Ullery.