Year: 2025
Type: article
Abstract: Coarse moduli spaces of Weierstrass fibrations over a smooth conic curve were constructed by the classical work of [Miranda] using geometric invariant theory. In our paper, we extend this treatment by using results of [Romagny] regarding group actions on stacks to give an explicit construction of the moduli stack $${\mathcal {W}}_{n}$$ W n of Weierstrass fibrations over a smooth conic curve with discriminant degree 12n and a section. We show that it is a smooth algebraic stack and prove that for $$n\ge 2$$ n ≥ 2 , the open substack $${\mathcal {W}}_{\textrm{min},n}$$ W min , n of minimal Weierstrass fibrations is a separated Deligne–Mumford stack over any base field K with $$\textrm{char}(K) \ne 2,3$$ char ( K ) ≠ 2 , 3 and $$\not \mid n$$ ∤ n . Arithmetically, for the moduli stack $${\mathcal {W}}_{\textrm{sf},n}$$ W sf , n of stable Weierstrass fibrations, we determine its motive in the Grothendieck ring of stacks to be $$\{{\mathcal {W}}_{\textrm{sf},n}\} = {\mathbb {L}}^{10n - 2}$$ { W sf , n } = L 10 n - 2 in the case that n is odd, which results in its weighted point count to be $$\#_q({\mathcal {W}}_{\textrm{sf},n}) = q^{10n - 2}$$ # q ( W sf , n ) = q 10 n - 2 over $${\mathbb {F}}_q$$ F q . In the appendix, we show how our methods can be applied similarly to the classical work of [Silverman] on coarse moduli spaces of self-maps of the projective line, allowing us to construct the natural moduli stack and to compute its motive.
Source: Mathematische Zeitschrift
Authors: Jun-Yong Park, Johannes Schmitt
Institutions: The University of Sydney, ETH Zurich
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W3184068299