Rational curves on del Pezzo surfaces in positive characteristic

Beheshti, Roya; Lehmann, Brian D.; Riedl, Eric; Tanimoto, Sho

doi:10.48550/arxiv.2110.00596

Cited by 2 publications

(1 citation statement)

References 57 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The reason for this is that we aim to emulate the Batyrev-Manin conjecture, and the form that conjecture should take for global fields of characteristic p is not fully settled. Indeed, there are counterexamples to the most naive formulations of Batyrev-Manin, even for the anticanonical height; see Starr-Tian-Zong [67, Lemma 5.1] and recent work of Beheshti, Lehmann, Riedl and Tanimoto [7].…”

Section: Weak Form Of the Stacky Batyrev-manin-malle Conjecturementioning

confidence: 99%

Heights on stacks and a generalized Batyrev–Manin–Malle conjecture

Ellenberg

Satriano

Zureick-Brown

2023

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $\mathcal {X}$ , which specializes to the Batyrev–Manin conjecture when $\mathcal {X}$ is a scheme and to Malle’s conjecture when $\mathcal {X}$ is the classifying stack of a finite group.

show abstract

Section: Weak Form Of the Stacky Batyrev-manin-malle Conjecturementioning

confidence: 99%