# An arithmetic count of the lines meeting four lines in $\mathbf {P}^3$

@article{Srinivasan2018AnAC,
title={An arithmetic count of the lines meeting four lines in \$\mathbf \{P\}^3\$},
author={P. Srinivasan and Kirsten Wickelgren},
journal={arXiv: Algebraic Geometry},
year={2018}
}
• Published 2018
• Mathematics
• arXiv: Algebraic Geometry
We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field $k$, this enrichment counts the number of lines meeting four lines defined over $k$ in $\mathbb{P}^3_k$, with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained… Expand
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#### References

SHOWING 1-10 OF 46 REFERENCES
An Arithmetic Count of the Lines on a Smooth Cubic Surface
• Mathematics
• 2017
We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over $\mathbb{C}$ there are $27$ lines, and over $\mathbb{R}$ the numberExpand
Toward an algebraic theory of Welschinger invariants
Let $S$ be a smooth del Pezzo surface over a field $k$ of characteristic $\neq 2, 3$. We define an invariant in the Grothendieck-Witt ring $GW(k)$ for "counting" rational curves in a curve class $D$Expand
Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry
• Mathematics
• 2002
Suppose that 2d - 2 tangent lines to the rational normal curve z → (1: z …: z d ) in d-dimensional complex projective space are given. It was known that the number of codimension 2 subspacesExpand
MOTIVIC EULER CHARACTERISTICS AND WITT-VALUED CHARACTERISTIC CLASSES
• M. Levine
• Mathematics
• Nagoya Mathematical Journal
• 2019
This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois.Expand
The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
• Mathematics
• 2005
We prove the B. and M. Shapiro conjecture that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomialsExpand
A classical proof that the algebraic homotopy class of a rational function is the residue pairing
• Mathematics
• 2016
Cazanave has identified the algebraic homotopy class of a rational function of $1$ variable with an explicit nondegenerate symmetric bilinear form. Here we show that Hurwitz's proof of a classicalExpand
Algebraic K-Theory and Quadratic Forms
The first section of this paper defines and studies a graded ring K . F associated to any field F. By definition, K~F is the target group of the universal n-linear function from F ~ x ... • F ~ to anExpand
Explicit Enumerative Geometry for the Real Grassmannian of Lines in Projective Space
We extend the classical Schubert calculus of enumerative geometry for the Grassmann variety of lines in projective space from the complex realm to the real. Specifically, given any collection ofExpand
Chow-Witt rings of Grassmannians
We complement our previous computation of the Chow-Witt rings of classifying spaces of special linear groups by an analogous computation for the general linear groups. This case involves discussionExpand
3264 and All That: A Second Course in Algebraic Geometry
• Mathematics
• 2016
Introduction 1. Introducing the Chow ring 2. First examples 3. Introduction to Grassmannians and lines in P3 4. Grassmannians in general 5. Chern classes 6. Lines on hypersurfaces 7. SingularExpand