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It is known that the monodromy group of each cover of a general curve of genus g>3 equals either the symmetric or the alternating group. The classical Catalan numbers count the minimal degree covers (with symmetric monodromy) of a general curve of even genus. We solve the analogous problem for the alternating group and we determine the number of alternating covers of minimal degree 2g+1 of a general curve of genus g., 18 pages. Minor improvements, also referencing the related work of Lian. Final version to appear in Annali Scuola Normale di Pisa

Using the connection discovered by Hassett between the Noether-Lefschetz moduli space of special cubic fourfolds of discriminant 42 and the moduli space F_{22} of polarized K3 surfaces of genus 22, we show that the universal K3 surface over F_{22} is unirational., 17 pages. Final version, to appear in Mathematische Annalen

We show that the canonical bundle of the Hurwitz stack classifying covers of genus g>1 and degree k>2 of the projective line is big. We show that all coarse moduli spaces of trigonal curves of genus g>1 are of general type., 16 pages. Final version, to appear in a special issue of Mathematische Zeitschrift in honor of Olivier Debarre

Using degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree $d$ and dimension $r$ on a general curve of genus $g$ satisfying prescribed incidence conditions at $n$ points. We determine these numbers completely for linear series of arbitrary dimension when $d$ is sufficiently large, and for all $d$ when either $r=1$ or $n=r+2$. Our formulas generalize and give new proofs of recent results of Tevelev and of Cela-Pandharipande-Schmitt., Comment: version 2, various corrections (including error in section 4), plus references to later work. To appear in J. Inst. Math. Jussieu

We prove two results. First, we establish that the normal bundle of any smooth curve of genus 7 having maximal Clifford index is stable. Note that 7 is the smallest genus for which such a result could possibly hold. We then show that rank four Lazarsfeld-Mukai vector bundles on a curve that lies on a general K3 surface are stable. Both results have consequences for Mercat's conjecture on higher rank vector bundles on generic curves.

The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in the moduli space of Deligne-Mumford stable pointed curves which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the moduli spaces of canonical divisors. In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus g curves are of pure codimension g in the moduli spaces of stable pointed curves. In addition to the closure of the canonical divisors on nonsingular curves, the moduli spaces have virtual components. In the Appendix, a complete proposal relating the sum of the fundamental classes of all components (with intrinsic multiplicities) to a formula of Pixton is proposed. The result is a precise and explicit conjecture in the tautological ring for the weighted fundamental class of the moduli spaces of twisted canonical divisors. As a consequence of the conjecture, the classes of the closures of the moduli spaces of canonical divisors on nonsingular curves are determined (both in the holomorphic and meromorphic cases)., Comment: Appendix on the weighted fundamental class by F. Janda, R. Pandharipande, A. Pixton, and D.Zvonkine [J. Institute Math. Jussieu, to appear]

We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K inside the second exterior product of a vector space, as well as a sharp upper bound for its Hilbert function. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, or the degree of growth and nilpotency class. For instance, we explicitly bound the aforementioned invariants in terms of the first Betti number for the maximal metabelian quotients of (1) the Torelli group associated to the moduli space of curves; (2) nilpotent fundamental groups of compact Kaehler manifolds; (3) the Torelli group of a free group., 25 pages. Final version, to appear in Duke Math. Journal

We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describe an explicit, characteristic-independent version of Hermite reciprocity for sl_2-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve., minor edits, 42 pages, to appear in Invent. Math

The Prym-Green conjecture predicts that the resolution of a general level p paracanonical curve of genus g is natural. Using decomposable ruled surfaces over an elliptic curve, we provide a complete solution (that is, for all levels) to this conjecture in odd genus., Comment: 15 pages. Final version, to appear in Geometry & Topology

These notes discuss recent advances on syzygies on algebraic curves, especially concerning the Green, the Prym-Green and the Green-Lazarsfeld Secant Conjectures. The methods used are largely geometric and variational, with a special emphasis on examples and explicit calculations. The notes are based on series of lectures given in Daejeon (March 2013), Rome (November–December 2015) and Guanajuato (February 2016).

We study the codimension two locus H in A_g consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class of H in A_g for every g. For g=4, this turns out to be the locus of Jacobians with a vanishing theta-null. For g=5, via the Prym map we show that H in A_5 has two components, both unirational, which we completely describe. This gives a geometric classification of 5-dimensional ppav whose theta-divisor has a quadratic singularity of non-maximal rank. We then determine the slope of the effective cone of A_5 and show that the component N_0' of the Andreotti-Mayer divisor has minimal slope 54/7. Furthermore, the Iitaka dimension of the linear system corresponding to N_0' is equal to zero.

We prove two statements concerning the linear strand of the minimal free resolution of a k-gonal curve C of genus g. Firstly, we show that a general curve C of genus g of non-maximal gonality k ≤ g + 1 2 satisfies Schreyer's Conjecture, that is, b g − k , 1 ( C , ω C ) = g − k . This statement goes beyond Green's Conjecture and predicts that all highest order linear syzygies in the canonical embedding of C are determined by the syzygies of the ( k − 1 ) -dimensional scroll containing C. Secondly, we prove an optimal effective version of the Gonality Conjecture for general k-gonal curves, which makes more precise the (asymptotic) Gonality Conjecture proved by Ein–Lazarsfeld and improves results of Rathmann.

We discuss, how via limit linear series and standard facts about divisors on moduli spaces of pointed curves, one can establish a non-existence Brill–Noether results for linear series with prescribed ramification at unassigned points.

We determine the Kodaira dimension of the Deligne–Mumford compactification \(\overline{\mathfrak{Diff }}_g\) of the universal difference variety over the moduli space of curves.

By analogy with Green's Conjecture on syzygies of canonical curves, the Prym-Green conjecture predicts that the resolution of a general level p paracanonical curve of genus g is natural. The Prym-Green Conjecture is known to hold in odd genus for almost all levels. Probabilistic arguments strongly suggested that the conjecture might fail for level 2 and genus 8 or 16. In this paper, we present three geometric proofs of the surprising failure of the Prym-Green Conjecture in genus 8, hoping that the methods introduced here will shed light on all the exceptions to the Prym-Green Conjecture for genera with high divisibility by 2., 23 pages; to appear in EPIGA Vol 1, paper no. 7

Using the isomorphism between the moduli space of polarized K3 surfaces of genus 14 and the moduli space of special cubic fourfolds of discriminant 26, we establish the rationality of the universal K3 surface of genus 14. Precisely, we show that the universal K3 surface of genus 14 is a projective bundle over a certain moduli space of nodal scrolls in P^5, whose rationality we prove using a degenerate version of Mukai's structure theorem for curves of genus 8., 20 pages. Final version, to appear in the Journal de Mathematiques Pures et Appliquees

We show that a general curve in an explicit class of what we call Du Val pointed curves satisfies the Brill-Noether Theorem for pointed curves. Furthermore, we prove that a generic pencil of Du Val pointed curves is disjoint from all Brill-Noether divisors on the universal curve. This provides explicit examples of smooth pointed curves of arbitrary genus defined over Q which are Brill-Noether general. A similar result is proved for 2-pointed curves as well using explicit curves on elliptic ruled surfaces., Comment: 14 pages, appeared in Selecta Mathematica. Slight inaccuracy in Section 2.1 corrected

Introduction.- Samuel Boissiere, Andrea Cattaneo, Marc Nieper-Wisskirchen, and Alessandra Sarti: The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface.- Igor Dolgachev: Orbital counting of curves on algebraic surfaces and sphere packings.- V. Gritsenko and K. Hulek: Moduli of polarized Enriques surfaces.- Brendan Hassett and Yuri Tschinkel: Extremal rays and automorphisms of holomorphic symplectic varieties.- Gert Heckman and Sander Rieken: An odd presentation for W(E_6).- S. Katz, A. Klemm, and R. Pandharipande, with an appendix by R. P. Thomas: On the motivic stable pairs invariants of K3 surfaces.- Shigeyuki Kondo: The Igusa quartic and Borcherds products.- Christian Liedtke: Lectures on supersingular K3 surfaces and the crystalline Torelli theorem.- Daisuke Matsushita: On deformations of Lagrangian fibrations.- G. Oberdieck and R. Pandharipande: Curve counting on K3 x E, the Igusa cusp form X_10, and descendent integration.- Keiji Oguiso: Simple abelian varieties and primitive automorphisms of null entropy of surfaces.- Ichiro Shimada: The automorphism groups of certain singular K3 surfaces and an Enriques surface.- Alessandro Verra: Geometry of genus 8 Nikulin surfaces and rationality of their moduli.- Claire Voisin: Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-Kahler varieties.

We formulate, and in some cases prove, three statements concerning the purity or, more generally, the naturality of the resolution of various modules one can attach to a generic curve of genus g and a torsion point of l in its Jacobian. These statements can be viewed an analogues of Green’s Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding non-vanishing locus in the moduli space $\mathcal{R}_{g,\ell}$ of twisted level l curves of genus g and use this to derive results about the birational geometry of $\mathcal{R}_{g, \ell}$ . For instance, we prove that $\mathcal{R}_{g,3}$ is a variety of general type when g>11 and the Kodaira dimension of $\mathcal{R}_{11,3}$ is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the Prym-Green conjecture in genus 8 and level 2.

We discuss various aspects of the geometry of theta characteristics including the birational geometry of the spin moduli space of curves, parametrization of moduli via special K3 surfaces, as well as the relation with classical theta function theory and string theory in the form of the superstring measure. A historical view of the development of the subject is also presented., Comment: 24 pages, final version. To appear in the Milan Journal of Mathematics

We study moduli spaces of K3 surfaces endowed with a Nikulin involution and their image in the moduli space R_g of Prym curves of genus g. We observe a striking analogy with Mukai's well-known work on ordinary K3 surfaces. Many of Mukai's results have a very precise Prym-Nikulin analogue, for instance: (1) A general Prym curve from R_g is a section of a Nikulin surface if and only if g\leq 7, g\neq 6. (2) R_7 has the structure of a Mori fibre space over the corresponding moduli space of polarized Nikulin surfaces. (3) In the case of genus next to maximal (g=6), the Prym-Nikulin locus in R_6 is an extremal divisor; the corresponding space of Nikulin surfaces has a Grassmannian GIT model. (4) The Prym-Nikulin locus in R_g can be characterized by extra syzygies of Prym-canonical curves. We then use these results to study the geometry of the moduli space S_g of even spin curves, with special emphasis on the transition case of S_8 which is a 21-dimensional Calabi-Yau variety., Comment: 26 pages. Final version to appear in Mathematische Annalen

We describe applications of Koszul cohomology to the Brill-Noether theory of rank 2 vector bundles. Among other things, we show that in every genus g>10, there exist curves invalidating Mercat's Conjecture for rank 2 bundles. On the other hand, we prove that Mercat's Conjecture holds for general curves of bounded genus, and its failure locus is a Koszul divisor in the moduli space of curves. We also formulate a conjecture concerning the minimality of Betti diagrams of suitably general curves, and point out its consequences to rank 2 Brill-Noether theory., 21 pages. Title slightly changed, various minor corrections. To appear in Pure and Applied Mathematics Quarterly, volume dedicated to Eckart Viehweg

The workshop on Moduli Spaces in Algebraic Geometry aimed to bring together researchers working in moduli theory, in order to discuss moduli spaces from different points of view, and to give an overview of methods used in their respective fields. Mathematics Subject Classification (2010): 14D22. Introduction by the Organisers The workshopModuli Spaces in Algebraic Geometry, organized by Dan Abramovich (Brown), Gavril Farkas (HU Berlin), Lucia Caporaso (Rome) and Stefan Kebekus (Freiburg) was held February 4–8, 2013 and was attended by 25 participants from around the world. The participants ranged from senior leaders in the field to young post-doctoral fellows and one advanced PhD student. The range of expertise covered areas ranging from classical algebraic geometry to mathematics inspired by string theory. Researchers reported on the substantial progress achieved within the last three years, discussed open problems, and exchanged methods and ideas. Most lectures were followed by lively discussions among participants, at times continuing well into the night. For a flavor of the range of subjects covered, a few of the talks are highlighted below. Stable pairs and knot invariants. Rahul Pandharipande (ETH Zurich) reported on work of Shende, Oblomkov and Maulik concerning Hilbert schemes Hilb(C, n) of n points on a curve C with an isolated, planar singularity. Building on ideas of Pandharipande-Thomas and Diaconescu, Shende and Oblomkov 344 Oberwolfach Report 06/2013 proposed a relation between the Euler characteristics χ ( Hilb(C, n) ) and coefficients in the HOMFLY polynomial of the curve singularity link. This was recently established by Maulik. Higher codimension loci in the moduli space of curves. Nicola Tarasca (Leibniz Universitat Hannover) reported on results in his PhD Thesis on the calculation of the cohomology class of the codimension two Brill-Noether locus of curves with a pencil of degree k in the moduli space M2k of stable curves of genus 2k. Remarkable here is that, while one has a large number of divisor class calculations on the moduli space, it is for the first time that a closed formula for a higher codimension locus on the moduli space is found. Tautological rings of moduli space of curves. In a very impressive talk, Aaron Pixton (Princeton) proposed a rather amazing conjecture generalizing at the level of the moduli space Mg,n the Faber-Zagier relations in the cohomology of the moduli space Mg. The increase in complexity when passing from smooth to singular curves is considerable and it is a major step forward that a concrete prediction has been put forward. The field is facing an interesting change of paradigm, in the sense that the largely accepted Faber Conjectures predicting that the corresponding tautological rings of moduli of curves satisfy Poincare duality, are being replaced by new predictions, according to which the suitable generalizations of Faber-Zagier relations span all relations between tautological classed. It is already clear that in genus 24 the two conjectures rule out each other (whereas for g < 24 they are equivalent) and it will be interesting to monitor future developments. Geometric compactifications of the moduli space of K3 surfaces. A classical unsolved problem of moduli theory asks for a modular compactification of the moduli space of polarized K3 surface. While several compactifications of the moduli space have been discussed, none of them is known to date to support a universal family. Bernd Siebert (Hamburg) reported on joint work Mark Gross (San Diego), Paul Hacking (Amherst) and Sean Keel (Austin) which might lead to a solution of this long-standing problem. Building on work of Gross-Siebert which uses Mirror symmetry to study degenerations of Calabi-Yau manifolds, there is hope to single out one particular toroidal compactification for which a family might exist. While many details still need to be filled in, and a discussion of the geometric and modular properties of the construction is still pending, this is a very exciting project which might eventually solve a classical problem. The moduli stack of semistable curves. This is a development providing a glimpse of the lively discussions which happened at this very meeting. Jarod Alper (ANU) reported in the most timely manner possible on present joint work with Andrew Kresch (Zurich) on the structure of the moduli stack of semistable curves. One of the main questions one must ask about any stack is whether or not it is a global-quotient stack, or at least if it can be approximated by a global-quotient stack. A central example is the stack M g of semistable curves, a keystone in constructing many moduli spaces. Kresch has shown that even the first stage Moduli Spaces in Algebraic Geometry 345 of this stack, M g , is not a global-quotient stack, but Alper conjectured that M g falls in a general class of stacks well-approximated by global-quotient stacks. Alper reported that this was established during this meeting by him and Kresch for M g , with strong evidence for the result to hold for the full moduli stack of semistable curves. Moduli of slope-semistable bundles. Daniel Greb (Ruhr-Universitat Bochum) reported on joint work with Matei Toma (Nancy), discussing wall-crossing and compactifications for moduli spaces of slope-semistable sheaves on higher-dimensional projective manifolds. Generalizing work of Joseph Le Potier and Jun Li, he constructed projective moduli spaces for slope-semistable sheaves by showing semiampleness of certain equivariant determinant line bundles. While the geometry of the resulting moduli spaces is presently only partially understood, these spaces are likely to shed new light on the question whether Tian’s topological compactifications of moduli spaces of slope-semistable vector bundles admit complex or even algebraic structures. Moduli Spaces in Algebraic Geometry 347 Workshop: Moduli Spaces in Algebraic Geometry

We determine the Kodaira dimension of the moduli space of even spin curves for all genera, with one possible exception: The scheme S_g has negative Kodaira dimension for g8. The Kodaira dimension of S_8 is non-negative., 9 pages. Improved presentation, to appear in Advances in Mathematics

For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. The aim of this paper is to completely clarify this problem in the case of the generic curve C of given genus. Using degeneration techniques and a few facts about the birational geometry of moduli spaces of stable pointed curves we determine precisely under which conditions the cycle of e-secant k-planes in non-empty and we compute its dimension. We also precisely determine the dimension of the variety of linear series on C carrying e-secant k-planes. In a different direction, in the last part of the paper we study the distribution of ramification points of the powers of a line bundle on C having prescribed ramification at a given point., Comment: 25 pages. Numerous changes suggested by the referee, several proofs explained in more detail. To appear in the Journal of the London Mathematical Society

Let $p_1,\dots, p_9$ be the points in $\mathbb A^2(\mathbb Q)\subset \mathbb P^2(\mathbb Q)$ with coordinates $$(-2,3),(-1,-4),(2,5),(4,9),(52,375), (5234, 37866),(8, -23), (43, 282), \Bigl(\frac{1}{4}, -\frac{33}{8} \Bigr)$$ respectively. We prove that, for any genus $g$, a plane curve of degree $3g$ having a $g$-tuple point at $p_1,\dots, p_8$, and a $(g-1)$-tuple point at $p_9$, and no other singularities, exists and is a Brill-Noether general curve of genus $g$, while a general curve in that $g$-dimensional linear system is a Brill-Noether-Petri general curve of genus $g$., New section added containing an explicit example of a 9-tuple of points in P^2(Q) that are of 3g-general for every g. Added a second proof of the fact that a du Val curve is BN general. Improved exposition

It has been a long-standing problem to find an adequate definition of a Clifford index for higher rank vector bundles on curves, which should capture the complexity of the curve in its moduli space. An interesting proposal in rank 2 has been put forward by Mercat, who conjectured that the second Clifford index of a curve should be equal to its classical Clifford index, defined in terms of gonality. Using moduli of sheaves on generic K3 surfaces, we prove Mercat's conjecture for generic curves of every genus. Furthermore, for odd g we identify an effective divisor in the moduli space M_g along which the Mercat Conjecture fails and we compute its slope, which is shown to be equal to 6+12/(g+1)., 14 pages. Final version, to appear in the American Journal of Mathematics

The Prym-Green Conjecture predicts that the resolution of a generic n-torsion paracanonical curve of every genus is natural. The conjecture has mostly been studied so far for level 2, that is, for Prym-canonical curves. Using a construction of Barth and Verra that realizes torsion bundles on sections of special K3 surfaces, we prove the Prym-Green Conjecture for curves of odd genus g and torsion bundles of sufficiently high order with respect to g. We also give partial results in even genus. In the process, we confirm the expectation of Barth and Verra concerning the number of curves in a fixed linear system on a K3 surface, having an n-torsion line bundle induced by restriction from the K3 surface., 15 pages, some results strengthened. Final version, to appear in Duke Math. Journal

The classical Castelnuovo numbers count linear series of minimal degree and fixed dimension on a general curve, in the case when this number is finite. For pencils, that is, linear series of dimension one, the Castelnuovo numbers specialize to the better known Catalan numbers. Using the Fulton-Pragacz determinantal formula for flag bundles and combinatorial manipulations, we obtain a compact formula for the number of linear series on a general curve having prescribed ramification at an arbitrary point, in the case when the expected number of such linear series is finite. The formula is then used to solve some enumerative problems on moduli spaces of curves., 10 pages. Final version, to appear in Mathematical Research Letters

We present a structure theorem for the moduli space R_7 of Prym curves of genus 7 as a projective bundle over the moduli space of 7-nodal rational curves. The existence of this parametrization implies the unirationality of R_7 and that of the moduli space of Nikulin surfaces of genus 7, as well as the rationality of the moduli space of Nikulin surfaces of genus 7 with a distinguished line. Using the results in genus 7, we then establish that R_8 is uniruled., Comment: 12 pages. To appear in Advances in Math

We compute the class of the compactification of the divisor of curves sitting on a K 3 K3 surface and show that it violates the Harris-Morrison Slope Conjecture. We carry this out using the fact that this divisor has four distinct incarnations as a geometric subvariety of the moduli space of curves. We also give a counterexample to a hypothesis raised by Harris and Morrison that the Brill-Noether divisors are essentially the only effective divisors on the moduli space of curves having minimal slope 6 + 12 / ( g + 1 ) 6+12/(g+1) .

We use geometrically defined divisors on moduli spaces of pointed curves to compute the graded Betti numbers of general sets of points on any nonhyperelliptic canonically embedded curve. This gives a positive answer to the Minimal Resolution Conjecture in the case of canonical curves. But we prove that the conjecture fails on curves of large degree. These results are related to the existence of theta divisors associated to certain stable vector bundles.

Generalizing the well-known Green Conjecture on syzygies of canonical curves, Green and Lazarsfeld formulated in 1986 the Secant Conjecture predicting that a line bundle L of sufficiently high degree on a curve has a non-linear p-syzygy if and only if L fails to be (p+1)-very ample. Via lattice theory for special K3 surfaces, Voisin's solution of the classical Green Conjecture and calculations on moduli stacks of pointed curves, we prove: (1) The Green-Lazarsfeld Secant Conjecture in various degree of generality, including its strongest possible form in the divisorial case in the universal Jacobian. (2) The Prym-Green Conjecture on the naturality of the resolution of a general Prym-canonical curve of odd genus., 24 pages. Final version, to appear in Inventiones Math

We establish a structure result for the universal abelian variety over the moduli space A_5, in terms of discriminant curves of conic bundles over a del Pezzo surface. In particular, this gives a very simple unirational parametrization of the boundary divisor of the moduli space A_6 of principally polarized abelian varieties of dimension 6. Using this we derive a lower bound on the slope of all effective divisors on A_6., Comment: 20 pages. Final version, to appear in Annales Scient. de l'ENS

We carry out a complete birational classification of the degree g universal Jacobian P_g over the moduli space of curves, highlighting the transition cases g=10, 11. The universal Jacobian is unirational when g11, the variety P_g has Kodaira dimension 3g-3, that is, the maximum allowed by Iitaka's easy addition formula for fibre spaces. In particular, we disprove the expectation that P_g and M_g have the same Kodaira dimension for all genera., 19 pages. Final version, to appear in Comm. Math. Helvetici

We describe the singular locus of the compactification of the moduli space $R_{g,l}$ of curves of genus $g$ paired with an $l$-torsion point in their Jacobian. Generalising previous work for $l\le 2$, we also describe the sublocus of noncanonical singularities for any positive integer $l$. For $g\ge 4$ and $l=3,4, 6$, this allows us to provide a lifting result on pluricanonical forms playing an essential role in the computation of the Kodaira dimension of $R_{g,l}$: for those values of $l$, every pluricanonical form on the smooth locus of the moduli space extends to a desingularisation of the compactified moduli space., Comment: 37 pages, 9 figures, to appear in J Eur Math Soc

We carry out a complete birational classification of the universal theta divisor Th_g over the moduli space of curves of genus g, and show that Th_g enjoys good rationality properties for g, 17 pages. Appeared in Journal de Mathematiques Pures et Appliquees. Typo corrected in statement of Theorem 5