Let q be a prime power, and let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$g\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In this article, we assume that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$G=\textrm{PSL}(5,\mathbb {F}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <mtext>PSL</mtext> <mo stretchy="false">(</mo> <mn>5</mn> <mo>,</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> acts on the projective points of the 4-dimensional projective geometry <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {P}^4_{\mathbb {F}_q}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mn>4</mn> </msubsup> </math></EquationSource> </InlineEquation>. We will show that G is not a genus g group if <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$q>3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we study the connectedness of the Hurwitz space <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {H}^{in}_{r}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mi>r</mi> <mrow> <mi mathvariant="italic">in</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for groups that possess genus zero and genus one.

On the realisability of $\textrm{PSL}(5,\mathbb {F}_q)$ as a low genus monodromy group

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2025年9月25日

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Amin M. Zebari,
Haval M. Mohammed Salih

摘要

Let q be a prime power, and let $g\le 1$ $g \leq 1$ . In this article, we assume that $G=\textrm{PSL}(5,\mathbb {F}_q)$ $G = PSL (5, F_{q})$ acts on the projective points of the 4-dimensional projective geometry $\mathbb {P}^4_{\mathbb {F}_q}$ $P_{F_{q}}^{4}$ . We will show that G is not a genus g group if $q>3$ $q > 3$ . Furthermore, we study the connectedness of the Hurwitz space $\mathcal {H}^{in}_{r}(G)$ $H_{r}^{in} (G)$ for groups that possess genus zero and genus one.

On the realisability of \(\textrm{PSL}(5,\mathbb {F}_q)\) as a low genus monodromy group

摘要