A modular compactification of ℳ1,n from A ∞-structures

We show that a certain moduli space of minimal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>A</m:mi> <m:mi>∞</m:mi> </m:msub> </m:math> A∞ -structures coincides with the modular compactification <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mover> <m:mi>ℳ</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {M̄}1,n(n-1) of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>ℳ</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> M1,n constructed by Smyth in [26]. In addition, we describe these moduli spaces and the universal curves over them by explicit equations, prove that they are normal and Gorenstein, show that their Picard groups have no torsion and that they have rational singularities if and only if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mn>11</m:mn> </m:mrow> </m:math> n≤ 11 .