Boundary complexes of moduli spaces of curves in higher genus

Emily Clader, Dante Luber, Kyla Quillin

Type: Article

Publication Date: 2020-12-16

Citations: 4

DOI: https://doi.org/10.1090/proc/15423

Abstract

Given a collection of boundary divisors in the moduli space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M overbar Subscript 0 comma n"> <mml:semantics> <mml:msub> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\overline {\mathcal {M}}_{0,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of stable genus-zero <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-pointed curves, Giansiracusa proved that their intersection is nonempty if and only if all pairwise intersections are nonempty. We give a complete classification of the pairs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis g comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(g,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the analogous statement holds in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M overbar Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\overline {\mathcal {M}}_{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

Proceedings of the American Mathematical Society - View

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