On the Kodaira dimension of Hurwitz spaces

The Hurwitz space H g is the parameter space of covers [f : C → P , p1, . . . , pb], whereC is a smooth algebraic curve of genus g and f is a degree k map simply branched over b = 2g + 2k − 2 distinct points p1, . . . , pb ∈ P . Note that we choose an ordering of the branch points of f . The origins of the interest in Hurwitz spaces go back to Riemann’s Existence Theorem and they have been used by Clebsch [Cl] and Hurwitz [Hu], as well as much later in [HM] to derive important information on the moduli space Mg of curves of genus g. We denote by Hg,k the moduli space of admissible covers constructed by Harris and Mumford [HM], whose study has been further refined in [ACV] via twisted stable maps. It comes equipped with two maps

[1]  G. Kempf,et al.  Instability in invariant theory , 1978, 1807.02890.

[2]  F. Cukierman.  Families of Weierstrass points , 1989 .

[3]  A. Hurwitz.  Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten , 1891 .

[4]  G. Geer,et al.  The Hodge Bundle on Hurwitz Spaces , 2010, 1010.3335.

[5]  D. Eisenbud,et al.  The Kodaira dimension of the moduli space of curves of genus ≧23 , 1987 .

[6]  Adam Logan,et al.  The Kodaira dimension of moduli spaces of curves with marked points , 2001, math/0110312.

[7]  G. Farkas,et al.  The universal theta divisor over the moduli space of curves , 2010, 1009.0184.

[8]  M. Fedorchuk,et al.  Toward GIT stability of syzygies of canonical curves , 2014, 1401.6101.

[9]  G. Farkas.  Koszul divisors on moduli spaces of curves , 2006, math/0607475.

[10]  M. Cornalba,et al.  Calculating cohomology groups of moduli spaces of curves via algebraic geometry , 1998, math/9803001.

[11]  M. Fedorchuk,et al.  Stability of 2nd Hilbert points of canonical curves , 2011, 1111.5339.

[12]  M. Fedorchuk.  Geometric invariant theory of syzygies, with applications to moduli spaces , 2017, 1712.02776.

[13]  B. Hassett,et al.  Log minimal model program for the moduli space of stable curves: The first flip , 2008, 0806.3444.

[14]  M. Fedorchuk,et al.  Finite Hilbert stability of (bi)canonical curves , 2011, 1109.4986.

[15]  A. Kneser.  Zur Theorie der algebraischen Functionen , 1887 .

[16]  Florian Geiss.  The unirationality of Hurwitz spaces of 6-gonal curves of small genus , 2011, Documenta Mathematica.

[17]  Karl-Otto Stöhr,et al.  Weierstrass gap sequences and moduli varieties of trigonal curves , 1992 .

[18]  Richárd Rimányi,et al.  Quadric ramk loci on moduli of curves and K3 surfaces , 2017, 1707.00756.

[19]  Mei-Chu Chang,et al.  The Kodaira dimension of the moduli space of curves of genus {15} , 1986 .

[20]  M. Cornalba,et al.  Footnotes to a paper of Beniamino Segre , 1981 .

[21]  Jacques Herbrand.  Zur Theorie der algebraischen Funktionen , 1932 .

[22]  Twisted Bundles and Admissible Covers , 2001, math/0106211.

[23]  D. Mumford,et al.  On the Kodaira dimension of the moduli space of curves , 1982 .

[24]  B. Hassett,et al.  Log canonical models for the moduli space of curves: The first divisorial contraction , 2006, math/0607477.

[25]  A. Kokotov,et al.  Isomonodromic tau function on the space of admissible covers , 2009 .