Moduli of Pointed Curves. G. Casnati, C. Fontanari

Moduli of Pointed Curves G. Casnati, C. Fontanari 1

1. Notation C is the field of complex numbers, GL k the general linear group of k k matrices with entries in C, P GL k the projective linear group, i.e. GL k modulo the subgroup of scalar matrices. If V is a vector space then P(V ) is the associated projective space. If V = C r+1 then P r := P(V ). A curve C is a projective scheme of dimension 1. If D is a divisor on C then ϕ D denotes the map induced on C by D. 2

M g,n will denote the coarse moduli space of smooth, projective n pointed curves (briefly n pointed curves in what follows) of genus g. A point of M g,n is then an ordered (n + 1) tuple (C, p 1,..., p n ), where C is a smooth and connected curve of genus g and p 1,..., p n C are pairwise distinct point. We set M g := M g,0. A general curve of genus g is a curve corresponding to the general point of the irreducible scheme M g. With abuse of notation, we will denote by C both the abstract curve and the corresponding closed point in M g. 3

2. Classical results on M g M 0 is a point, M 1 = A 1 the identification being given by the j invariant. More in general for each g 2 the scheme M g has dimension 3g 3. If g = 0, 1 then M g is rational, i.e. its points are in a birational correspondence with the points of a projective space of suitable dimension. In 1960 J. Igusa proved the rationality of M 2. It is then natural to ask the following Question 1 Is M g rational for all g 2? 4

The problem of checking the rationality of a scheme X is in general quite hard. It is often easier to check its unirationality, i.e. the existence of a dominant rational map from a projective space onto X. If dim(x) 2 it is classically known that rationality and unirationality are equivalent notions. If dim(x) 3 V. Iskovskih and J. Manin proved that this is no more true. Using suitable plane models F. Severi proved in [13] that M g is unirational for g 10 and conjectured Conjecture 2 M g is unirational for all g. Severi s proof has been analized and demonstrated in a rigorous form by E. Arbarello and E. Sernesi. 5

3. Kodaira dimension of M g The scheme M g is irreducible but (unfortunately) it is not projective but only quasi projective. It is then natural to introduce the so called Deligne Mumford compactification M g (when g 2). Its points represent isomorphism classes of nodal curves with finite automorphism group. It is possible to define the Kodaira dimension κ g of M g (or M g ). In the three papers [8], [10] and [11] D. Eisenbud, J. Harris and D. Mumford proved the following Theorem 3 If g 24 then κ g = 3g 3. Moreover κ 23 1. thus showing that Severi s conjecture is false. 6

More recently G. Farkas improved the above result when g = 23 (see [9]) proving Theorem 4 κ 23 2. Moreover he conjectures that equality actually holds. He also asserts in his home page that M 22 is of general type. It is thus natural to ask Question 5 For which g is M g unirational? 7

4. Unirationality of M g The first new result (after the one by Igusa) is due to E. Sernesi who proved in [13] the unirationality of M 12. Its proof has been then generalized by M. Chang and Z. Ran in [5] covering also the unirationality of M g when g = 11, 13. While the cases g 10 rest on the existence of a suitable plane model, when g 11 such a description is no more possible. In these cases it is necessary to work with space models in P 3. Such curves are not in general complete intersection in P 3, but they are the zero locus of global sections of suitable vector bundles. Since a monad for such bundles can be described and unirationality follows from such a description. 8

In the papers [6] and [7] M. Chang and Z. Ran also proved κ 15, κ 16 = : in these cases it is not known if M g is actually unirational. More recently unirationality has been proved also for M 14 by A. Verra in [15]. No results are known in the range 17 g 21(22) but J. Harris and D. Morrison made the following Conjecture 6 κ g = for g 22. 9

5. Rationality of M g The rationality of M g were proved by P.I. Katsylo and N.I. Shepherd Barron in the ten years from 1987 till 1996 for g = 3, 4, 5, 6. All such results are based on the properties of the canonical models of curves of such genera. Such models are either complete intersection in the canonical space (g = 3, 4, 5) or on a fixed del Pezzo surface of degree 5 (g = 6). When g 7 there is not such an easy and useful description. Surprisingly the case g = 3 has revealed to be the harder one, though the description is the easiest. Indeed the general curve of genus g = 3 can be canonically embedded as smooth plane quartic, hence a model of M 3 can be obtained by taking the quotient of the space of plane quartics modulo the natural action of P GL 3. 10

6. M g,n and M g,n An n pointed curve is an isomorphism class of ordered (n+1) tuples (C, P 1,..., P n ) where C M g and P 1,..., P n C are pairwise distinct points. A stable n pointed curve is an isomorphism class of ordered (n + 1) tuples (C, P 1,..., P n ) where C is nodal of arithmetic genus g, the points P 1,..., P n C are pairwise distinct and the automorphism group of the (n + 1) tuple is finite. In the construction of the compactification M g appears naturally the moduli spaces of n pointed curves M g,n and its compactification via stable n pointed curves M g,n. Indeed each component of := M g \ M g is obtained via identification of pairs of points from moduli spaces of strictly smaller genus. 11

7. The Kodaira dimension of M g,n. As for unpointed curves one can ask for informations about M g,n and its compactification M g,n. Since there exists a natural forgetting morphism M g,n M g whose fibre over the general curve C is the n th product of C we immediately infer that M g,n is irreducible and its dimension is 3g 3 + n. It is then natural to look for an analogous of Theorem 3 regarding the Kodaira dimension κ g,n of M g,n. In the paper [12], A. Logan asserts that Theorem 7 If g 23 then κ g,n = 3g 3 + n. 12

A. Logan proves that M g,n has canonical singularities if g 4. Thus each pluricanonical divisor on the open set of curves without non trivial automorphisms inside M g,n extends to a pluricanonical divisor on the desingularization of M g,n. This fact allows him to prove the following improvement of Theorem 7. Theorem 8 If 4 g 22 then κ g,n 0 for n h(g) and κ g,n = 3g 3 + n for n k(g) where g = 4 5 6 7 8 9 h(g) = 16 15 16 14 14 13 k(g) = 17 16 17 15 14 13 g = 10 11 12 13 14 15 h(g) = 14 11 13 11 11 10 k(g) = 15 12 13 11 11 11 g = 16 17 18 19 20 21 22 h(g) = 11 9 9 8 7 5 3 k(g) = 12 9 10 8 7 6 4 Moreover except in the cases g = 16, 18, 22 then κ g,h(g) 1. 13

There is no evidence that the above theorem is sharp. We recall the above mentioned, perhaps conjectural, Farkas assertion on κ 22, which would imply that M 22,n is of general type for all n (same argument used for proving Theorem 7). When g = 23 then κ 23,n 2 for all n and the unique result in Logan s paper is the following Theorem 9 All but finitely many M 23,n are of general type. When g = 2, 3 Logan s argument cannot be applied directly, since in these two cases the singularities of the scheme M g,n are not necessarily canonical. The only result is Theorem 10 If g = 2, 3 all but finitely many M g,n dominate a variety of general type. 14

8. Unirationality of M g,n Due to the results listed in the previous section, in view of the facts that κ g = for g 16 and that M g is unirational (and sometimes rational) for g 14, it is natural to ask the following natural Question 11 What can be said about M g,n when n < h(g)? Is κ g = in that range? Is it unirational (or rational)? It is evident that M 0,n is always rational. In his Ph.D. Thesis [3] P. Belorousski proved the rationality of M 1,n when n 10. In [4] G. Bini and C. Fontanari prove that κ 1,11 = 0 and κ 1,n = 1 for n 11. 15

In the last section of [12], A. Logan tries to generalize such results, checking at least for g 11 when his main Theorem 8 is sharp. In particular he asserts the following Theorem 12 If 2 g 11, g 10, then M g,n is unirational for n f(g) where g = 2 3 4 5 6 7 8 9 11 f(g) = 11 14 15 12 15 11 5 8 10 Since we recall that g = 4 5 6 7 8 9 11 h(g) = 16 15 16 14 14 13 11 Theorem 8 is sharp at least in the three cases g = 4, 6, 11. 16

The proof of the above result rests on the existence of particular models for general curves of given genus. For g = 2, A. Logan considers curves of bidegree 82, 3) on a smooth quadric surface. For g = 3, 4, 5, 6, he considers the canonical models. For g = 7, he considers plane curves of degree 7 with 8 nodes. For g = 8, 9, he considers plane curves of degree 7 with 13, 12 nodes. For g = 11, he considers hyperplane sections of a K 3 surface. Using the description in [1] due to E. Arbarello and M. Cornalba one can improve Logan s bound obtaining f(8) = 8 and f(10) = 3. 17

9. Rationality of M g,n In view of the rationality of M g for 2 g 6, it is natural to inspect more deeply the above unirationality results. In our paper we prove the following Theorem 13 If 2 g 6 then M g,n is rational for n r(g) where g = 2 3 4 5 6 r(g) = 12 14 15 12 15 Notice that r(2) = f(2) + 1 and r(g) = f(g) for 3 g 6. 18

The proof of the above theorem for g 3 rests on the following construction. If (C, p 1,..., p g 3 ) is a general pointed curve of genus g 3, then K p 1 p g 3 induces a map ϕ: C C K P 2 coinciding with the projection of its canonical model from the points p 1,..., p 3. It is not difficult to see that ϕ is birational. C K is a (g+1) tic passing through the points A i := ϕ(p i ) which are in general position in the plane. 19

One of the key points of our proof is the following Lemma 14 Let 3 g 6. Then C K is an integral (g+1) tic with δ := (g 2 3g)/2 nodes N 1,..., N δ. There is a unique smooth (g 3) tic E C containing the points N 1,..., N δ, A 1,..., A g 3. If g 5 the points N 1,..., N δ, A 1, A 2 uniquely determine A 3,..., A g 3. Conversely, each integral (g + 1) tic D P 2 with exactly δ := (g 2 3g)/2 nodes N 1,..., N δ lying on an integral smooth (g 3) tic E is obtained in the above way. 20

Thus we can associate to (C, p 1,..., p g 3 ) the plane (g +1) tic D with δ nodes and g 3 distinguished points A 1,..., A g 3 on an integral (g 3) tic E. It is easy to prove that two such plane data correspond to the same pointed curve if and only if they are projectively equivalent. For example let us examine the case g = 4. In this case K P 1 is a plane quintic with two nodes N 1, N 2 and a distinguished point A 1. If we choose coordinates in P 2 in such a way that N 1 := [1, 0, 0], N 2 := [0, 1, 0], A 1 := [0, 0, 1] the linear system Σ of such curves is a P 14. 21

Thus we can consider the following incidence variety X n { (D, A 2,..., A n ) Σ (P 2 ) n 1 A i D }. If n 15 the natural projection X n (P 2 ) n 1 endows X with a structure of projective bundle with fibre P 15 n. M g,n is birational to X/P GL 2 and via standard results one then checks that this last quotient is rational. 22

References [1] E. Arbarello, M. Cornalba Footnotes to a paper of Beniamino Segre. Math. Ann. 256 (1981), no. 3, 341 362. [2] E. Arbarello, E. Sernesi The equation of a plane curve. Duke Math. J. 46 (1979), no. 2, 469 485. [3]P. Belorousski Chow rings of moduli spaces of pointed elliptic curves. Ph.D. Thesis (1998). [4] G. Bini, C. Fontanari Moduli of curves and spin structures via algebraic geometry, Trans. Amer. Math. Soc. 358 (2006), no. 7, 3207 3217. [5] M.C. Chang, Z. Ran Unirationality of the moduli spaces of curves of genus 11, 13 (and 12), Invent. Math. 76 (1984), no. 1, 41 54. 23

[6] M.C. Chang, Z. Ran The Kodaira dimension of the moduli space of curves of genus 15. J. Differential Geom. 24 (1986), no. 2, 205 220. [7] M.C. Chang, Z. Ran On the slope and Kodaira dimension of M g for small g. J. Differential Geom. 34 (1991), no. 1, 267 274. [8] D. Eisenbud, J. Harris The Kodaira dimension of the moduli space of curves of genus 23. Invent. Math. 90 (1987), no. 2, 359 387. [9] G. Farkas The geometry of the moduli space of curves of genus 23. Math. Ann. 318 (2000), no. 1, 43 65. [10] J. Harris, D. Mumford On the Kodaira dimension of the moduli space of curves. With an appendix by William Fulton. Invent. Math. 67 (1982), no. 1, 23 88. 24

[11] J. Harris On the Kodaira dimension of the moduli space of curves. II. The even-genus case. Invent. Math. 75 (1984), no. 3, 437 466. [12] A. Logan The Kodaira dimension of moduli spaces of curves with marked points. Amer. J. Math. 125 (2003), no. 1, 105 138. [13] E. Sernesi L unirazionalitá della varietá dei moduli delle curve di genere 12. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 3, 405 439. [14] F. Severi Sulla classificazione delle curve algebriche e sul teorema di esistenza di Riemann. Rend. R. Acc. Naz. Lincei (5) 24 (1921), 877 888. [15] A. Verra The unirationality of the moduli spaces of curves of genus 14 or lower. Compos. Math. 141 (2005), no. 6, 1425 1444. 25