arXiv subjects

Eric Riedl

Publications and source records attributed to Eric Riedl.

33 records · Page 2Linked to original sources

arXiv2021

Clustered families and applications to Lang-type conjectures

We introduce and classify 1-clustered families of linear spaces in the Grassmannian $\mathbb{G}(k-1,n)$ and give applications to Lang-type conjectures. Let $X \subset \mathbb{P}^n$ be a very general hypersurface of degree $d$. Let $Z_L$ be the locus of points contained in a line of $X$. Let $Z_2$ be the locus of points on $X$ that are swept out by lines that meet $X$ in at most $2$ points. We prove that 1) If $d \geq \frac{3n+2}{2}$, then $X$ is algebraically hyperbolic outside $Z_L$. 2) If $d \geq \frac{3n}{2}$, $X$ contains lines but no other rational curves 3) If $d \geq \frac{3n+3}{2}$, then the only points on $X$ that are rationally Chow zero equivalent to points other than themselves are contained in $Z_2$. 4) If $d \geq \frac{3n+2}{2}$ and a relative Green-Griffiths-Lang Conjecture holds, then the exceptional locus for $X$ is contained in $Z_2$.

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arXiv2021

Restricted tangent bundles for general free rational curves

Suppose that $X$ is a smooth projective variety and that $C$ is a general member of a family of free rational curves on $X$. We prove several statements showing that the Harder-Narasimhan filtration of $T_{X}|_{C}$ is approximately the same as the restriction of the Harder-Narasimhan filtration of $T_{X}$ with respect to the class of $C$. When $X$ is a Fano variety, we prove that the set of all restricted tangent bundles for general free rational curves is controlled by a finite set of data. We then apply our work to analyze Peyre's "freeness" formulation of Manin's Conjecture in the setting of rational curves.

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arXiv2020

Linear subspaces of hypersurfaces

Let $X$ be an arbitrary smooth hypersurface in $\mathbb{C} \mathbb{P}^n$ of degree $d$. We prove the de Jong-Debarre Conjecture for $n \geq 2d-4$: the space of lines in $X$ has dimension $2n-d-3$. We also prove an analogous result for $k$-planes: if $n \geq 2 \binom{d+k-1}{k} + k$, then the space of $k$-planes on $X$ will be irreducible of the expected dimension. As applications, we prove that an arbitrary smooth hypersurface satisfying $n \geq 2^{d!}$ is unirational, and we prove that the space of degree $e$ curves on $X$ will be irreducible of the expected dimension provided that $d \leq \frac{e+n}{e+1}$.

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arXiv2020

Positivity Results for spaces of rational curves

Let $X$ be a very general hypersurface of degree $d$ in $\mathbb{P}^n$. We investigate positivity properties of the spaces $R_e(X)$ of degree $e$ rational curves in $X$. We show that for small $e$, $R_e(X)$ has no rational curves meeting the locus of smooth embedded curves. We show that for $n \leq d$, there are no rational curves in the locus $Y \subset X$ swept out by lines. And we exhibit differential forms on a smooth compactification of $R_e(X)$ for every $e$ and $n-2 \geq d \geq \frac{n+1}{2}$.

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arXiv2019

Algebraic hyperbolicity of very general surfaces

Recently, Haase and Ilten initiated the study of classifying algebraically hyperbolic surfaces in toric threefolds. We complete this classification for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, $\mathbb{P}^2 \times \mathbb{P}^1$, $\mathbb{F}_e \times \mathbb{P}^1$ and the blowup of $\mathbb{P}^3$ at a point, augmenting our earlier work on $\mathbb{P}^3$. In the process, we codify several different techniques for proving algebraic hyperbolicity, allowing us to prove similar results for hypersurface in any variety admitting a group action with dense orbit.

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arXiv2019

Effective bounds on ampleness of cotangent bundles

We prove that a general complete intersection of dimension $n$, codimension $c$ and type $d_1, \dots, d_c$ in $\mathbb{P}^N$ has ample cotangent bundle if $c \geq 2n-2$ and the $d_i$'s are all greater than a bound that is $O(1)$ in $N$ and quadratic in $n$. This degree bound substantially improves the currently best-known super-exponential bound in $N$ by Deng, although our result does not address the case $n \leq c < 2n-2$.

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arXiv2018

Applications of a grassmannian technique in hypersurfaces

In this paper we further develop a Grassmannian technique used to prove results about very general hypersurfaces and present three applications. First, we provide a short proof of the Kobayashi Conjecture given previous results on the Green-Griffiths-Lang Conjecture. Second, we characterize the dimension of the space of Chow-equivalent points on a very general hypersurface, proving the remaining cases of a conjecture of Chen, Lewis and Sheng and providing a short, alternate proof for many of the already known cases. Finally, we relate Seshadri constants of very general points to Seshadri constants of arbitrary points of very general hypersurfaces.

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arXiv2018

Algebraic hyperbolicity of the very general quintic surface in $\mathbb{P}^3$

We prove that a curve of degree $dk$ on a very general surface of degree $d \geq 5$ in $\mathbb{P}^3$ has geometric genus at least $\frac{dk(d-5)+k}{2} + 1$. This improves bounds given by G. Xu. As a corollary, we conclude that the very general quintic surface in $\mathbb{P}^3$ is algebraically hyperbolic.

math.AG↗

arXiv2018

Nef cones of Hilbert schemes of points on surfaces

Let X be a smooth projective surface of irregularity 0. The Hilbert scheme of n points on X parameterizes zero-dimensional subschemes of X of length n. In this paper, we discuss general methods for studying the cone of ample divisors on the Hilbert scheme. We then use these techniques to compute the cone of ample divisors on the Hilbert scheme for several surfaces where the cone was previously unknown. Our examples include families of surfaces of general type and del Pezzo surfaces of degree 1. The methods rely on Bridgeland stability and the Positivity Lemma of Bayer and Macri.

math.AG↗

arXiv2017

Normal bundles of rational curves in projective space

Let $b_{\bullet}$ be a sequence of integers $1 < b_1 \leq b_2 \leq \cdots \leq b_{n-1}$. Let $M(b_{\bullet})$ be the space parameterizing nondegenerate, rational curves of degree $e$ in $\mathbb{P}^n$ with ordinary singularities such that the normal bundle has the splitting type $\bigoplus_{i=1}^{n-1}\mathcal{O}(e+b_i)$. When $n=3$, celebrated results of Eisenbud, Van de Ven, Ghione and Sacchiero show that $M(b_{\bullet})$ is irreducible of the expected dimension. We show that when $n \geq 5$, these loci are generally reducible with components of higher than the expected dimension. We give examples where the number of components grows linearly with $n$. These generalize an example of Alzati and Re.

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arXiv2017

Normal bundles of rational curves on complete intersections

Let $X \subset \mathbb{P}^n$ be a general Fano complete intersection of type $(d_1,\dots, d_k)$. If at least one $d_i$ is greater than $2$, we show that $X$ contains rational curves of degree $e \leq n$ with balanced normal bundle. If all $d_i$ are $2$ and $n\geq 2k+1$, we show that $X$ contains rational curves of degree $e \leq n-1$ with balanced normal bundle. As an application, we prove a stronger version of the theorem of Z. Tian \cite{Tian}, Q. Chen and Y. Zhu \cite{ChenZhu} that $X$ is separably rationally connected by exhibiting very free rational curves in $X$ of optimal degrees.

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arXiv2016

Kontsevich spaces of rational curves on Fano hypersurfaces

We investigate the spaces of rational curves on a general hypersurface. In particular, we show that for a general degree $d$ hypersurface in $\mathbb{P}^n$ with $n \geq d+2$, the space $\overline{\mathcal{M}_{0,0}}(X,e)$ of degree $e$ Kontsevich stable maps from a rational curve to $X$ is an irreducible local complete intersection stack of dimension $e(n-d+1)+n-4$.

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arXiv2016

Rational curves on complete intersections in positive characteristic

We study properties of rational curves on complete intersections in positive characteristic. It has long been known that in characteristic 0, smooth Calabi-Yau and general type varieties are not uniruled. In positive characteristic, however, there are well-known counterexamples to this statement. We will show that nevertheless, a \emph{general} Calabi-Yau or general type complete intersection in projective space is not uniruled. We will also show that the space of complete intersections of degree $(d_1, \cdots, d_k)$ containing a rational curve has codimension at least $\sum_{i=1}^k d_i - 2n + 2$ in the moduli space of all complete intersections of given multidegree and dimension.

math.AG↗

arXiv2016

Rational curves on general type hypersurfaces

We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an application, we use a result of Clemens and Ran to prove that a very general hypersurface of degree (3n+1)/2 \leq d \leq 2n-3 contains lines but no other rational curves.

math.AG↗

arXiv2014

Gröbner Bases and Nullstellensätze for Graph-Coloring Ideals

We revisit a well-known family of polynomial ideals encoding the problem of graph-$k$-colorability. Our paper describes how the inherent combinatorial structure of the ideals implies several interesting algebraic properties. Specifically, we provide lower bounds on the difficulty of computing Gröbner bases and Nullstellensatz certificates for the coloring ideals of general graphs. For chordal graphs, however, we explicitly describe a Gröbner basis for the coloring ideal, and provide a polynomial-time algorithm.

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