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Brian Lehmann

Publications and source records attributed to Brian Lehmann.

At least 19 recordsLinked to original sources
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.

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arXiv2026

Manin's conjecture for semi-integral curves and $\mathbb A^1$-connectedness

We explore log Manin's conjecture for integral points and its connections to $\mathbb A^1$-connectedness. We prove log Manin's conjecture for Campana rational curves and for $\mathbb A^1$-curves on split toric varieties. Our arguments combine the Cox ring description of the moduli space of rational curves with Batyrev's heuristic-type counting arguments. As our proofs are geometric in nature, they give a geometric explanation of the mysterious leading constant for Campana points proposed by Chow--Loughran--Takloo-Bighash--Tanimoto.

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arXiv2026

Geometric Manin's conjecture in characteristic $p$

Geometric Manin's conjecture for complex Fano varieties describes the structure of the moduli space of curves. We propose a version of this conjecture in characteristic $p$ and describe its connection to the Batyrev--Manin--Peyre--Tschinkel conjecture over global fields. This is a survey paper written for a volume of the Summer Research Institute in Algebraic Geometry held at Colorado State University in 2025.

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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.

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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.

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arXiv2025

Homological stability and Manin's conjecture for rational curves on quartic del Pezzo surfaces

We prove a version of Manin's conjecture (over $\mathbb{F}_{q}$ for $q$ large) and the Cohen--Jones--Segal conjecture (over $\mathbb{C}$) for maps from rational curves to split quartic del Pezzo surfaces. The proofs share a common method which builds upon prior work of the first and fourth authors. The main ingredients of this method are (i) the construction of bar complexes formalizing the inclusion-exclusion principle and its point counting estimates, (ii) dimension estimates for spaces of rational curves using conic bundle structures, (iii) estimates of error terms using arguments of Sawin--Shusterman based on Katz's results, and (iv) a certain virtual height zeta function revealing the compatibility of bar complexes and Peyre's constant. Our argument substantiates the heuristic approach to Manin's conjecture over global function fields given by Batyrev and Ellenberg--Venkatesh in this case.

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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.

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arXiv2024

Campana rational connectedness and weak approximation

Campana introduced a notion of Campana rational connectedness for Campana orbifolds. Given a Campana fibration over a complex curve, we prove that a version of weak approximation for Campana sections holds at places of good reduction when the general fiber satisfies a slightly stronger version of Campana rational connectedness. Campana also conjectured that any Fano orbifold is Campana rationally connected; we verify a stronger statement for toric Campana orbifolds. A key tool in our study is log geometry and moduli stacks of stable log maps.

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arXiv2024

Approximating rational points on surfaces

Let $X$ be a smooth projective algebraic variety over a number field $k$ and $P$ in $X(k)$. In 2007, the second author conjectured that, in a precise sense, if rational points on $X$ are dense enough, then the best rational approximations to $P$ must lie on a curve. We present a strategy for deducing a slightly weaker conjecture from Vojta's conjecture, and execute this strategy for the full conjecture for split surfaces.

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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 $\mathcal{X} \to B$.

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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 $\mathbb{P}^n$.

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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.

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arXiv2022

Classifying sections of del Pezzo fibrations, I

We develop a strategy to classify the components of the space of sections of a del Pezzo fibration over $\mathbb{P}^{1}$. In particular, we prove the Movable Bend and Break lemma for del Pezzo fibrations. Our approach is motivated by Geometric Manin's Conjecture and proves upper bounds on the associated counting function. We also give applications to enumerativity of Gromov-Witten invariants and to the study of the Abel-Jacobi map.

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arXiv2022

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.

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arXiv2022

Moduli spaces of rational curves on Fano threefolds

We prove several classification results for the components of the moduli space of rational curves on a smooth Fano threefold. In particular, we prove a conjecture of Batyrev on the growth of the number of components as the degree increases. The key to our approach is Geometric Manin's Conjecture which predicts the number of components parameterizing free curves.

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arXiv2022

Geometric consistency of Manin's Conjecture

We conjecture that the exceptional set in Manin's Conjecture has an explicit geometric description. Our proposal includes the rational point contributions from any generically finite map with larger geometric invariants. We prove that this set is contained in a thin subset of rational points, verifying there is no counterexample to Manin's Conjecture which arises from an incompatibility of geometric invariants.

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arXiv2021

Classifying sections of del Pezzo fibrations, II

Let $X$ be a del Pezzo surface over the function field of a complex curve. We study the behavior of rational points on $X$ leading to bounds on the counting function in Geometric Manin's Conjecture. A key tool is the Movable Bend and Break Lemma which yields an inductive approach to classifying relatively free sections for a del Pezzo fibration over a curve. Using this lemma we prove Geometric Manin's Conjecture for certain split del Pezzo surfaces of degree $\geq 2$ admitting a birational morphism to $\mathbb P^2$ over the ground field.

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