<p>In a previous article we showed how the line pairs that occur as asymptotes to hyperbolas in a pencil of affine conics can be detected from their incidence geometry alone, without reference to the pencil. In this article we classify the configurations of all such line pairs up to affine equivalence. Many of our methods work over an arbitrary field <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1237_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation> of characteristic <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1237_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$\ne 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≠</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and we use tools from the theory of algebraic curves and projective duality to obtain a complete classification of the collections of line pairs that are asymptotic to a pencil of conics if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1237_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation> is real closed or algebraically closed, while we obtain a partial classification if <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1237_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation> is a finite field. A classification for other fields remains an open question. Ultimately this is a question regarding affine equivalence within a system of certain rational quartic curves.</p>

Geometry of lines asymptotic to a pencil of conics

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发表日期：

2025年10月22日

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Bruce Olberding,
Elaine A. Walker

摘要

In a previous article we showed how the line pairs that occur as asymptotes to hyperbolas in a pencil of affine conics can be detected from their incidence geometry alone, without reference to the pencil. In this article we classify the configurations of all such line pairs up to affine equivalence. Many of our methods work over an arbitrary field $\mathbb {k}$ $k$ of characteristic $\ne 2$ $\neq 2$ , and we use tools from the theory of algebraic curves and projective duality to obtain a complete classification of the collections of line pairs that are asymptotic to a pencil of conics if $\mathbb {k}$ $k$ is real closed or algebraically closed, while we obtain a partial classification if $\mathbb {k}$ $k$ is a finite field. A classification for other fields remains an open question. Ultimately this is a question regarding affine equivalence within a system of certain rational quartic curves.