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>
- 1 Department of Mathematics and Computer Science, University of Perugia, Perugia, Italy
Abstract
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> </mrow> </math> . Then a maximal bundle of 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> having in common one directrix cubic curve is constructed.
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