Mathematical Research Letters

Volume 20 (2013)

Number 5

Sharp slope bounds for sweeping families of trigonal curves

Pages: 869 – 884



Anand Deopurkar (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Anand Patel (Department of Mathematics, Boston College, Chestnut Hill, Massachusetts, U.S.A.)


We establish sharp bounds for the slopes of curves in $\overline{M}_g$ that sweep out the locus of trigonal curves, reproving Stankova-Frenkel’s bound of $7 + 6/g$ for even $g$ and obtaining the bound $7 + 20 / (3g + 1)$ for odd $g$. For even $g$, we find an explicit expression of the so-called Maroni divisor in the Picard group of the space of admissible triple covers. For odd $g$, we describe the analogous extremal effective divisor and give a similar explicit expression.

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