Conics meeting eight lines over perfect fields

Cameron Darwin; Aygul Galimova; Miao Pam Gu; Stephen McKean

doi:10.1016/j.jalgebra.2023.04.024

Home/Archive/arXiv

arXiv · 2107.05543

Conics meeting eight lines over perfect fields

Cameron Darwin·Aygul Galimova·Miao Pam Gu·Stephen McKean

View original source ↗Read source document ↗Export metadata ↓

Abstract

Over the complex numbers, there are 92 plane conics meeting 8 general lines in projective 3-space. Using the Euler class and local degree from motivic homotopy theory, we give an enriched version of this result over any perfect field. This provides a weighted count of the number of plane conics meeting 8 general lines, where the weight of each conic is determined the geometry of its intersections with the 8 given lines. As a corollary, real conics meeting 8 general lines come in two families of equal size.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Cameron Darwin, Aygul Galimova, Miao Pam Gu, Stephen McKean. 2023-04-25. Conics meeting eight lines over perfect fields. https://doi.org/10.1016/j.jalgebra.2023.04.024

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

K-stable Fano 3-folds in the families 2.18 and 3.4

We prove that smooth Fano 3-folds in the families 2.18 and 3.4 are K-stable.

math.AG↗

arXiv2026

G3-Criteria and Applications

The G3-property of a subvariety was introduced by Hironaka-Matsumura, and plays an important role for deducing connectedness and extension results. Unfortunately, it's a rather elusive notion, which is not always easy to establish. Most of the existing work is concentrated on subvarieties of homogeneous varieties. The first goal of this article is to show that mobility assumptions on the subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a certain partial positivity property, slightly stronger than G3, previously introduced by the author. Second, we apply the result to prove that, in numerous situations, the splitting of the normal bundle of a smooth two-codimensional subvariety implies that it is a complete intersection.

math.AG↗

arXiv2026

Nodal degeneration of chiral algebras I: Global structure and gluing formula

We define a natural extension of a universal factorization algebra $\mathcal{A}$ to families of stable punctured curves, by integrating over all semistable modifications. We prove that the resulting sheaf of factorization homology satisfies a natural gluing formula, by tensoring over a certain derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, generalizing the Verlinde formula for gluing of conformal blocks.

math.AG↗