For a set <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {L}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation> of lines of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$\text{ PG }(n,q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.333333em" /> <mtext>PG</mtext> <mspace width="0.333333em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, a set X of points of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$\text{ PG }(n,q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.333333em" /> <mtext>PG</mtext> <mspace width="0.333333em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is called an <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {L}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-blocking set if each line of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {L}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation> contains at least one point of X. Consider a possibly singular quadric Q of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$\text{ PG }(n,q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.333333em" /> <mtext>PG</mtext> <mspace width="0.333333em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and denote by <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {S}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation> (respectively, <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {T}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">T</mi> </math></EquationSource> </InlineEquation>) the set of all lines of <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$\text{ PG }(n,q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.333333em" /> <mtext>PG</mtext> <mspace width="0.333333em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> meeting Q in 2 (respectively, 1 or <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq12.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$q+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) points. For <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq13.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="119" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {L}}\in \{{\mathcal {S}},{\mathcal {T}}\cup {\mathcal {S}}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mi mathvariant="script">S</mi> <mo>,</mo> <mi mathvariant="script">T</mi> <mo>∪</mo> <mi mathvariant="script">S</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, we find the minimal cardinality of an <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {L}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-blocking set of <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$\text{ PG }(n,q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.333333em" /> <mtext>PG</mtext> <mspace width="0.333333em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and determine all <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2024_1559_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {L}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-blocking sets of that minimal cardinality.

Blocking sets of secant and tangent lines with respect to a quadric of $\text{ PG }(n,q)$

期刊论文
发表日期：

2025年1月17日

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Bart De Bruyn,
Puspendu Pradhan,
Binod Kumar Sahoo

摘要

For a set ${\mathcal {L}}$ $L$ of lines of $\text{ PG }(n,q)$ $PG (n, q)$ , a set X of points of $\text{ PG }(n,q)$ $PG (n, q)$ is called an ${\mathcal {L}}$ $L$ -blocking set if each line of ${\mathcal {L}}$ $L$ contains at least one point of X. Consider a possibly singular quadric Q of $\text{ PG }(n,q)$ $PG (n, q)$ and denote by ${\mathcal {S}}$ $S$ (respectively, ${\mathcal {T}}$ $T$ ) the set of all lines of $\text{ PG }(n,q)$ $PG (n, q)$ meeting Q in 2 (respectively, 1 or $q+1$ $q + 1$ ) points. For ${\mathcal {L}}\in \{{\mathcal {S}},{\mathcal {T}}\cup {\mathcal {S}}\}$ $L \in {S, T \cup S}$ , we find the minimal cardinality of an ${\mathcal {L}}$ $L$ -blocking set of $\text{ PG }(n,q)$ $PG (n, q)$ and determine all ${\mathcal {L}}$ $L$ -blocking sets of that minimal cardinality.

Blocking sets of secant and tangent lines with respect to a quadric of \(\text{ PG }(n,q)\)

摘要