%0 Journal Article
%T Subplanes of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow>  <mi>P</mi><mi>G</mi><mrow><mo>(</mo>   <mrow>    <mn>2,</mn><msup>     <mi>q</mi>     <mn>3</mn>    </msup>    </mrow>  <mo>)</mo></mrow></mrow></math> and the Ruled Varieties <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow>  <msubsup>   <mi>V</mi>   <mn>2</mn>   <mn>5</mn>  </msubsup>  </mrow></math> of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <mrow>  <mi>P</mi><mi>G</mi><mrow><mo>(</mo>   <mrow>    <mn>6,</mn><mi>q</mi></mrow>  <mo>)</mo></mrow></mrow></math>
%A Rita Vincenti
%J Open Journal of Discrete Mathematics
%P 16-27
%@ 2161-7643
%D 2024
%I Scientific Research Publishing
%R 10.4236/ojdm.2024.142003
%X In this note we study subplanes of order <i>q</i> of the projective plane <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>Π</mi><mo>=</mo><mi>P</mi><mi>G</mi><mrow><mo>(</mo>
   <mrow>
    <mn>2,</mn><msup>
     <mi>q</mi>
     <mn>3</mn>
    </msup>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>
 and the ruled varieties <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>V</mi>
   <mn>2</mn>
   <mn>5</mn>
  </msubsup>
  </mrow>
</math>
 of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>Σ</mi><mo>=</mo><mi>P</mi><mi>G</mi><mrow><mo>(</mo>
   <mrow>
    <mn>6,</mn><mi>q</mi></mrow>
  <mo>)</mo></mrow></mrow>
</math>
 using the spatial representation of Π in Σ, by fixing a hyperplane <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <msup>
  <mi>Σ</mi>
  <mo>′</mo>
 </msup>
</math>
 with a regular spread of planes. First are shown some configurations of the affine <i>q</i>-subplanes. Then to prove that a variety <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>V</mi>
   <mn>2</mn>
   <mn>5</mn>
  </msubsup>
  </mrow>
</math>
 of Σ represents a non-affine subplane of order <i>q</i> of Π, after having shown basic incidence properties of it, such a variety <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>V</mi>
   <mn>2</mn>
   <mn>5</mn>
  </msubsup>
  </mrow>
</math>
 is constructed by choosing appropriately the two directrix curves in two complementary subspaces of Σ. The result can be translated into further incidence properties of the affine points of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>V</mi>
   <mn>2</mn>
   <mn>5</mn>
  </msubsup>
%K Finite Geometry
%K Translation Planes
%K Spreads
%K Varieties
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=132725