Sharp slope bounds for sweeping families of trigonal curves

Abstract

Abstract. We establish sharp bounds for the slopes of curves in 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.

The Cycle Classes of Divisorial Maroni Loci

“…We do not know whether m min coincides with the closure of the Maroni locus, but in the trigonal case (d = 3 and g even) it does: the class of the closure of the Maroni locus coincides with m L,N for suitable (L, N ). In fact, in [5] Deopurkar and Patel determined the class of the Zariski closure of the Maroni divisor in the trigonal case and we show that an appropriate choice of L and N reproduces the class found by Deopurkar and Patel. We also show that the effective class found by Patel in [17] containing the Maroni locus on a partial compactification is in general larger than the class of the Zariski closure of the Maroni locus. We point out that we do not use the standard tool of test curves that seems inadequate here.…”

“…In this section we compare our results with the results of Patel [17] on a partial compactification and with the results of Deopurkar-Patel [5] for the trigonal case.…”

“…Patel determined classes in a partial compactification of the Hurwitz space. Deopurkar and Patel calculated the class of the closure of the Maroni divisor in the trigonal case in [5].…”

Syzygy divisors on Hurwitz spaces

“…It would also be interesting to find replacements for µ i when d does not divide g − 1. This would be analogous to the replacement of the Maroni divisor in the case of odd genus trigonal curves found in [6].…”

Stability Conditions and Positivity of Invariants of Fibrations

“…It is widely believed (see for instance [7], [29]) that there should exist a lower bound for the slope of fibred surfaces increasing with the gonality of the general fibres (under some genericity assumption). This conjecture, however, is only proved for some step: hyperelliptic fibrations (the slope inequality), trigonal fibrations [7], [19] and fibrations with general gonality [29], [20]. Recently Beorchia and Zucconi [12] have proved some results also on fourgonal fibred surfaces.…”