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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

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 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.

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 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

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 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.

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

We describe the birational geometry of the moduli space S_g^{-} of odd spin curves (theta-characteristics) for all genera g. The odd spin moduli space is a uniruled variety for g, 34 pages. Final version, to appear in the Annals of Mathematics

We study the enumerative geometry of the moduli space R_g of Prym varieties of dimension g-1 (also known as the space of admissible double covers). Our main result is that the compactification of R_g is of general type as soon as g>13. We achieve this by computing the class of two types of cycles on R_g: one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym-Green conjecture on syzygies of Prym-canonical curves. In the appendix we show that even though R_g has non-canonical singularities, pluricanonical forms on R_g extend to any desingularization., 36 pages. Corrected typo in a formula. Appeared in the Journal of the European Mathematical Society

We perform an intersection theoretic study of the rational map between two different moduli spaces of stable curves which associates to a curve its corresponding Brill-Noether locus (in the case this locus has virtual dimension 1). We then use these results to describe the cone of moving divisors on M_g. Several other applications to moduli spaces of Prym varieties are presented. In a different direction, we prove that the locus in M_g of curves failing to satisfy the Gieseker-Petri theorem is supported in codimension 1 for every possible type of linear series., 34 pages. Minor corrections. To appear in the Journal of Algebraic Geometry

Given a moduli problem with a nice coarse moduli space, which is the most intrinsic and natural divisor on this moduli space? We describe a general method of constructing effective divisors on a large class of moduli spaces using the syzygies of the parametrized objects. In this paper we treat the case of the moduli space of curves and we compute the slope of a doubly infinite sequence of Koszul divisors on M_g. We present a single formula which apart from giving myriads of new counterexamples to the Harris-Morrison Slope Conjecture it also encodes virtually all interesting divisor class calculations on M_g. In particular our formula specializes to earlier results due to Harris-Mumford (the case of the Brill-Noether divisors), Eisenbud-Harris (Giseker-Petri divisors), Khosla as well as to our earlier counterexamples to the Slope Conjecture obtained using K3 surfaces and Hurwitz schemes. We also describe five ways of constructing Koszul divisors on moduli spaces of pointed curves. As an application, we improve most of Logan's results on the Kodaira type of M_{g,n}., 40 pages, minor corrections. To appear in the American Journal of Mathematics

We analyze the stratification of the moduli space S_g of spin curves of genus g given by the dimension of the theta-characteristic. Using the relation between gaussian maps and the strata S_g^r, we construct "regular" components of S_g^r having expected codimension r(r+1)/2 inside S_g. We also relate moduli spaces of pointed curves with a moving spin structure to the classical Gieseker-Petri loci in M_g. We show that the locus of curves for which the Gieseker-Petri theorem fails for a pencil is always a divisor on M_g. Finally, we give a sufficient criterion for the injectivity of Gaussian maps of arbitrary line bundles on general curves of genus g., Comment: 22 pages. Minor revisions, to appear in Crelle

We compute the Mori cone of curves of the moduli space \M_{g,n} of stable n-pointed curves of genus g in the case when g and n are relatively small. For instance, we show that for g, 16 pages, references added

A component of the moduli space M_g(Y,b) of stable maps from genus g curves to a variety Y is said to be regular if it is generically smooth and of the expected dimension provided by deformation theory. In this note we prove existence of regular components of M_g(Y,b) when Y is a product of projective spaces. This result can also be seen as a general position theorem for Brill-Noether loci in M_g., 11 pages, final version

We prove that the Kodaira dimension of the moduli space M_{23} of curves of genus 23 is at least 2. We also present some evidence for the hypothesis that the Kodaira dimension of the moduli space is actually equal to 2. Note that for g > 23 the moduli space is of general type, while for g\leq 22, Harris and Morrison conjectured that M_g is uniruled. The result on M_{23} is obtained by investigating the relative position of three explicit multicanonical divisors which are of Brill-Noether type., 21 pages, Latex

We discuss the role of K3 surfaces in the context of Mercat's conjecture in higher rank Brill–Noether theory. Using liftings of Koszul classes, we show that Mercat's conjecture in rank 2 fails for any number of sections and for any gonality stratum along a Noether–Lefschetz divisor inside the locus of curves lying on K3 surfaces. Then we show that Mercat's conjecture in rank 3 fails even for curves lying on K3 surfaces with Picard number 1. Finally, we provide a detailed proof of Mercat's conjecture in rank 2 for general curves of genus 11, and describe explicitly the action of the Fourier–Mukai involution on the moduli space of curves.