Searcharxiv⌕ Search

arXiv subjects

Eric Riedl

Publications and source records attributed to Eric Riedl.

At least 19 recordsLinked to original sources
arXiv2026

Nonexistence of degree two rational multisections of conic bundles over the plane

We prove that a standard conic bundle $X \to \mathbb P^2_{\mathbb C}$ whose discriminant is very general of degree at least 18 admits no rational multisections of degree two. This is the first step towards proving a conjecture of Iskovskikh that there are conic bundle threefolds that are not unirational, since to prove that $X$ is not unirational, it suffices to show that there are no rational multisections of any degree. Proving Iskovskikh's conjecture would provide the first example of a rationally connected variety that is not unirational.

math.AG↗
arXiv2026

Finding large families of rational curves through Bend-and-Break

We present a new construction that allows us to break off large-degree rational curves from families of higher genus curves. Our construction and results deepen the connection between rational curves and positivity of the anticanonical divisor. Specifically, we show that varieties with large Fujita invariant admit large families of rational curves. We also construct free rational curves on certain singular Fano varieties. As an explicit consequence of our results, we prove that for a general Fano hypersurface of index at least 3, all spaces of genus g curves of sufficiently large degree have the expected dimension.

math.AG↗
arXiv2026

Computing Jet Differentials and the Green-Griffiths-Lang Conjecture for Complements of Smooth Plane Curves

We study the Green-Griffiths-Lang Conjecture for complements of smooth plane curves. We develop an effective method for computing a family of negatively twisted invariant logarithmic 2-jet differentials. By realizing the first logarithmic jet space as a hypersurface in $\mathbb{P}^2 \times \mathbb{P}^2$, we encode these jet differentials in a finitely generated bigraded module that can be computed explicitly. We use this description to give a computational criterion for the Green-Griffiths-Lang Conjecture and verify it for several families of smooth plane curves. In examples with sufficiently many independent jet differentials, we determine the exceptional locus explicitly.

math.AG↗
arXiv2025

Separable rational connectedness and $k$-plane sections of hypersurfaces

Let X be a smooth hypersurface of degree d in P^n over an algebraically closed field of characteristic p. We show that X must be separably rationally connected and must contain a free line if either p is at least d or if p is at least d-1 and the defining equation has some partial derivative that is not too singular. We also show that X must be separably rationally connected in any characteristic if d = 4 and n is sufficiently large. Along the way, we generalize results on the spaces of k-planes in X to characteristic p and connect some of these questions to the spaces of linear sections of X.

math.AG↗
arXiv2025

Free curves and fundamental groups

We show that klt Fano varieties and certain lc Fano varieties contain free higher-genus curves in their smooth loci. Our methods also allow us to find free curves on varieties in positive characteristic and on quasiprojective varieties, under a natural positivity condition on the tangent bundle. We then use the existence of free curves to deduce finiteness of the fundamental group of the smooth locus in these settings. The paper includes an appendix by de Jong that establishes the K\"unneth formula for tame \'etale fundamental groups.

math.AG↗
arXiv2025

Optimal bounds in Bend-and-Break

We improve the Bend-and-Break result of Miyaoka and Mori by establishing the optimal degree bound. Our result also yields optimal bounds on lengths of extremal rays of log canonical pairs.

math.AG↗
arXiv2024

A transfer principle for unirationality

We apply ideas related to the strength of polynomials to provide new cases of unirational hypersurfaces. It is famously known that hypersurfaces that are smooth in very high codimension are unirational, and a simple corollary then implies that any polynomial of sufficiently high strength will give rise to a unirational hypersurface. Our main result shows that unirationality is preserved under a substitution of high collective strength. In particular, we prove that polynomials of sufficiently high secondary strength are unirational. Along the way, we introduce a ``transfer principle,'' showing that polynomials of high collective strength have Fano schemes defined by polynomials of high collective strength. This gives an alternate proof of a result of Xi Chen on unirationality of Fano schemes, and proves a weakened form of the de Jong-Debarre Conjecture. Combined with some ideas of Starr, this implies a version of Kazhdan and Ziegler's result about the universality of complete intersections of polynomials.

math.AG↗
arXiv2024

Codimension of jumping loci

Suppose that $\mathcal{E}$ is a vector bundle on a smooth projective variety $X$. Given a family of curves $C$ on $X$, we study how the Harder-Narasimhan filtration of $\mathcal{E}|_{C}$ changes as we vary $C$ in our family. Heuristically we expect that the locus where the slopes in the Harder-Narasimhan filtration jump by $\mu$ should have codimension which depends linearly on $\mu$. We identify the geometric properties which determine whether or not this expected behavior holds. We then apply our results to study rank $2$ bundles on $\mathbb{P}^{2}$ and to study singular loci of moduli spaces of curves.

math.AG↗
arXiv2023

On the asymptotic enumerativity property for Fano manifolds

We study the enumerativity of Gromov-Witten invariants where the domain curve is fixed in moduli and required to pass through the maximum possible number of points. We say a Fano manifold satisfies asymptotic enumerativity if such invariants are enumerative whenever the degree of the curve is sufficiently large. Lian and Pandharipande speculate that every Fano manifold satisfies asymptotic enumerativity. We give the first counterexamples, as well as some new examples where asymptotic enumerativity holds. The negative examples include special hypersurfaces of low Fano index and certain projective bundles, and the new positive examples include many Fano threefolds and all smooth hypersurfaces of degree $d \leq (n+3)/3$ in $\mathbb{P}^n$.

math.AG↗
arXiv2023

Non-free curves on Fano varieties

Let $X$ be a smooth Fano variety over $\mathbb{C}$ and let $B$ be a smooth projective curve over $\mathbb{C}$. Geometric Manin's Conjecture predicts the structure of the irreducible components $M \subset \mathrm{Mor}(B, X)$ parametrizing curves which are non-free and have large anticanonical degree. Following ideas of our previous work, we prove the first prediction of Geometric Manin's Conjecture describing such irreducible components. As an application, we prove that there is a proper closed subset $V \subset X$ such that all non-dominant components of $\mathrm{Mor}(B, X)$ parametrize curves in $V$, verifying an expectation put forward by Victor Batyrev. We also demonstrate two important ways that studying $\mathrm{Mor}(B,X)$ differs from studying the space of sections of a Fano fibration $\mathcal{X} \to B$.

math.AG↗
arXiv2023

Non-free sections of Fano fibrations

Let $B$ be a smooth projective curve and let $\pi: \mathcal{X} \to B$ be a smooth integral model of a geometrically integral Fano variety over $K(B)$. Geometric Manin's Conjecture predicts the structure of the irreducible components $M \subset \textrm{Sec}(\mathcal{X}/B)$ which parametrize non-relatively free sections of sufficiently large anticanonical degree. Over the complex numbers, we prove that for any such component $M$ the sections come from morphisms $f: \mathcal{Y} \to \mathcal{X}$ such that the generic fiber of $\mathcal{Y}$ has Fujita invariant $\geq 1$. Furthermore, we prove that there is a bounded family of morphisms $f$ which together account for all such components $M$. These results verify the first part of Batyrev's heuristics for Geometric Manin's Conjecture over $\mathbb{C}$. Our result has ramifications for Manin's Conjecture over global function fields: if we start with a Fano fibration over a number field and reduce mod $p$, we obtain upper bounds of the desired form by first letting the prime go to infinity, then the height.

math.AG↗
arXiv2022

Algebraic Hyperbolicity of Complements of Generic Hypersurfaces in Projective Spaces

We study the algebraic hyperbolicity of the complement of very general degree $2n$ hypersurfaces in P^n. We prove the Algebraic Green-Griffiths-Lang Conjecture for these complements, and in the case of the complement of a quartic plane curve, we completely characterize the exceptional locus as the union of the flex and bitangent lines.

math.AG↗
arXiv2022

Lines highly tangent to a hypersurface

We study spaces of lines that meet a smooth hypersurface X in P^n to high order. As an application, we give a polynomial upper bound on the number of planes contained in a smooth degree d hypersurface in P^5 and provide a proof of a result of Landsberg without using moving frames.

math.AG↗
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 $Z_2$.

math.AG↗
arXiv2021

Rational curves on del Pezzo surfaces in positive characteristic

We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic 0. We also investigate the principles of Geometric Manin's Conjecture for weak del Pezzo surfaces. In the course of this investigation, we give examples of weak del Pezzo surfaces defined over $\mathbb{F}_{2}(t)$ or $\mathbb{F}_{3}(t)$ such that the exceptional sets in Manin's Conjecture are Zariski dense.

math.AG↗
arXiv2021

Restrictions on rational surfaces lying in very general hypersurfaces

We study rational surfaces on very general Fano hypersurfaces in $\mathbb{P}^n$, with an eye toward unirationality. We prove that given any fixed family of rational surfaces, a very general hypersurface of degree $d$ sufficiently close to $n$ and $n$ sufficiently large will admit no maps from surfaces in that family. In particular, this shows that for such hypersurfaces, any rational curve in the space of rational curves must meet the boundary. We also prove that for any fixed ratio $\alpha$, a very general hypersurface in $\mathbb{P}^n$ of degree $d$ sufficiently close to $n$ will admit no maps from a surface satisfying $H^2 \geq \alpha HK$, where $H$ is the pullback of the hyperplane class from $\mathbb{P}^n$ and $K$ is the canonical bundle on the surface.

math.AG↗
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.

math.AG↗