<p>Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\overline{{\mathcal {M}}_{0,5}^{\mathbb {R}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <msubsup> <mi mathvariant="script">M</mi> <mrow> <mn>0</mn> <mo>,</mo> <mn>5</mn> </mrow> <mi mathvariant="double-struck">R</mi> </msubsup> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation> be the Deligne–Mumford compactification of the moduli space of genus 0 real algebraic curves with five marked points. By <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathcal {L}}(\overline{{\mathcal {M}}_{0,5}^{\mathbb {R}}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mover> <msubsup> <mi mathvariant="script">M</mi> <mrow> <mn>0</mn> <mo>,</mo> <mn>5</mn> </mrow> <mi mathvariant="double-struck">R</mi> </msubsup> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> we denote its orientation cover. The cell decomposition of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathcal {L}}(\overline{{\mathcal {M}}_{0,5}^{\mathbb {R}}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mover> <msubsup> <mi mathvariant="script">M</mi> <mrow> <mn>0</mn> <mo>,</mo> <mn>5</mn> </mrow> <mi mathvariant="double-struck">R</mi> </msubsup> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a dessin d’enfant of genus 4. In this paper, we compute the Belyi pair for this dessin. In particular, it turns out that the corresponding curve is the celebrated Bring curve.</p>

BELYI PAIR FOR THE ORIENTATION COVER OF $\varvec{\overline{\mathcal {M}_{0,5}^{\mathbb {R}}}}$

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发表日期：

2026年1月1日

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N. Ya Amburg,
E. M. Kreines

摘要

Let $\overline{{\mathcal {M}}_{0,5}^{\mathbb {R}}}$ $\bar{M_{0, 5}^{R}}$ be the Deligne–Mumford compactification of the moduli space of genus 0 real algebraic curves with five marked points. By ${\mathcal {L}}(\overline{{\mathcal {M}}_{0,5}^{\mathbb {R}}})$ $L (\bar{M_{0, 5}^{R}})$ we denote its orientation cover. The cell decomposition of ${\mathcal {L}}(\overline{{\mathcal {M}}_{0,5}^{\mathbb {R}}})$ $L (\bar{M_{0, 5}^{R}})$ is a dessin d’enfant of genus 4. In this paper, we compute the Belyi pair for this dessin. In particular, it turns out that the corresponding curve is the celebrated Bring curve.

BELYI PAIR FOR THE ORIENTATION COVER OF \(\varvec{\overline{\mathcal {M}_{0,5}^{\mathbb {R}}}}\)

摘要