The divisor D 2 d

Remember that PicQ(Mg,1) is generated by the Hodge class λ, the cotangent class ψ corresponding to the marked point, and the boundary classes δ₀, . . . δ_g−1 defined as follows. The classδ0 is the class of the closure of the locus of pointed irreducible nodal curves, and the class δ_i is the class of the closure of the locus of pointed curves [Ci ∪ C_g−i, p] whereC_i andC_g−i are smooth curves respectively of genusiand g−imeeting transversally in one point, and p is a smooth point in C_i, for i = 1, . . . , g −1. In this section we prove the following theorem.

Theorem 3.2.1. Let g = 3s and d= 2s+ 2 for s≥1. The class of the divisor D²_d in PicQ(Mg,1) is

h

D²_dⁱ=aλ+cψ−

g−1X

i=0

b_iδ_i

3.2 The divisor D²_d

where

a= 48s⁴+ 80s³−16s²−64s+ 24 (3s−1)(3s−2)(s+ 3) N_g,2,d c= 2s(s−1)

3s−1 N_g,2,d

b₀ = 24s⁴+ 23s³−18s²−11s+ 6 3(3s−1)(3s−2)(s+ 3) N_g,2,d b₁ = 14s³+ 6s²−8s

(3s−2)(s+ 3)N_g,2,d b_g−1 = 48s⁴+ 12s³−56s²+ 20s

(3s−1)(3s−2)(s+ 3) N_g,2,d. Moreover for g= 6 and for i= 2,3,4,we have that

b_i = −7i²+ 43i−6.

3.2.1 The coefficient c

The coefficientccan be quickly found. LetCbe a general curve of genusg and consider the curve C = {[C, y] : y ∈ C} in Mg,1 obtained varying the point y on C. Then the only generator class having non-zero intersection withC isψ, andC·ψ= 2g−2. On the other hand,C·D²_dis equal to the number of triples (x, y, l)∈C×C×G²_d(C) such thatx and y are different points andh⁰(l(−x−y))≥2. The number of such linear series on a generalC is computed by the Castelnuovo number (remember thatρ = 0), and for each of them the number of couples (x, y) imposing only one condition is twice the number of double points, computed by the Plücker formula. Hence we get the equation

D²_d·C= 2

(d−1)(d−2)

2 −g

N_g,2,d =c(2g−2) and so

c= 2s(s−1) 3s−1 N_g,2,d. 3.2.2 The coefficients a and b₀

In order to compute a and b₀, we use a Porteous-style argument. Let G_d² be the family parametrizing triples (C, p, l), where [C, p]∈ M^irr_g,1 andlis ag²_donC; denote byη: G_d² → M^irr_g,1 the natural map. There existsπ:Y_d² → G_d² a universal pointed quasi-stable curve, with σ: G_d² → Y_d² the marked section. Let L → Y_d² be the universal line bundle of relative degree dtogether with the trivialization σ^∗(L) ∼=O_G2

d, and V ⊂π_∗(L) be the sub-bundle which over each point (C, p, l = (L, V)) in G_d² restricts to V. (See [Kho07,

§2] for more details.)

Furthermore let us denote byZ_d² the family parametrizing ((C, p), x1, x₂, l),

where [C, p]∈ M^irr_g,1,x₁, x₂ ∈C andl is ag²_d on C, and letµ, ν:Z_d² → Y_d² be defined as the maps that send ((C, p), x1, x₂, l) respectively to ((C, p), x1, l) and ((C, p), x2, l).

Now given a linear series l= (L, V), the natural map ϕ:V →H⁰(L|p+x) globalizes to

ϕ:e V →µ_∗(ν^∗L ⊗O/I_Γ

σ+∆) =:M

as a map of vector bundles over Y_d², where ∆ and Γ_σ are the loci in Z_d² determined respectively byx₁ =x₂ and x₂ =p. ThenD²_d∩ M^irr_g,1 is the push-forward of the locus in Y_d² where ϕe has rank≤1. Using Porteous formula, we have

h D²_dⁱ

M^irr_g,1 =η_∗π_∗ V^∨

M^∨

2

(3.1)

=η_∗π_∗ π^∗c₂(V^∨) +π^∗c₁(V^∨)·c₁(M) +c²₁(M)−c₂(M).

Let us find the Chern classes ofM. Tensoring the exact sequence 0→I_∆/I_∆+Γ

σ →O/I_∆+Γ

σ →O_∆→0

byν^∗L and applying µ_∗, we deduce that ch(M) =ch(µ∗(O_Γ

σ(−∆)⊗ν^∗L)) +ch(µ∗(O_∆⊗ν^∗L))

=ch(µ_∗(O_Γ

σ(−∆))) +ch(µ_∗(O_∆⊗ν^∗L))

=e^−σ+ch(L) hence

c₁(M) =c₁(L)−σ c₂(M) =−σc₁(L).

The following classes

α=π_∗c₁(L)²∩^hY_d²ⁱ γ =c₁(V)∩^hG_d²ⁱ

3.2 The divisor D²_d

have been studied in [Kho07, Thm. 2.11]. In particular 6(g−1)(g−2)

dN_g,2,d η_∗(α)|_M^irr

g,1 = 6(gd−2g²+ 8d−8g+ 4)λ + (2g²−gd+ 3g−4d−2)δ₀

−6d(g−2)ψ, 2(g−1)(g−2)

N_g,2,d η_∗(γ)|_M^irr

g,1 = (−(g+ 3)ξ+ 40)λ +1

6((g+ 1)ξ−24)δ₀

−3d(g−2)ψ, where

ξ = 3(g−1) +(g+ 3)(3g−2d−1) g−d+ 5 . Plugging into (3.1) and using the projection formula, we find

h D²_dⁱ

M^irr_g,1 =η_∗−γ·π_∗c₁(L) +γ·π_∗σ+α+π_∗σ²−π_∗(σc1(L))

= (1−d)η∗(γ) +η∗(α)−N_g,2,d·ψ.

Hence

a= 48s⁴+ 80s³−16s²−64s+ 24 (3s−1)(3s−2)(s+ 3) N_g,2,d b₀ = 24s⁴+ 23s³−18s²−11s+ 6

3(3s−1)(3s−2)(s+ 3) N_g,2,d and we recover the previously computed coefficient c.

3.2.3 The coefficient b₁

Let C be a general curve of genusg−1 and (E, p, q) a two-pointed elliptic curve, with p−q not a torsion point in Pic⁰(E). Let C1 := {(C ∪y∼q E, p)}y∈C be the family of curves obtained identifying the pointq ∈E with a moving pointy∈C. Computing the intersection of the divisorD²_dwithC₁ is equivalent to answering the following question:

how many triples (x, y, l) are there, withy ∈C,x ∈C∪y∼qE\ {p} and l ={lC, l_E} a limitg²_d onC∪y∼qE, such that (p, x, l) arises as limit of (pt, xt, lt) on a family of curves {Ct}t with smooth general element, where p_t and x_t impose only one condition on l_t a g²_d?

Let a^lE(q) = (a0, a₁, a₂) be the vanishing sequence of l_E ∈ G²_d(E) at q. Since C is general, there are nog²_d−1onC, hencel_C is base-point free anda₂ =d. Moreover we know a₁ ≤d−2. Let us supposex∈E\{q}. We distinguish two cases. Ifρ(E, q) =ρ(C, y) = 0, then w^lE(q) = ρ(1,2, d) = 3d−8. Thus a^lE(q) = (d−3, d−2, d). Removing the base

point we have that lE(−(d−3)q) is a g²₃ and lE(−(d−3)q−p−x) produces a g¹₁ on E, hence a contradiction. The other case is ρ(E, q) = 1 andρ(C, y)≤ −1. These force a^lE(q) = (d−4, d−2, d) and a^lC(y)≥(0,2,4). OnEwe have thatl_E(−(d−4)q−p−x) is ag¹₂.

The question splits in two: firstly, how many linear seriesl_E ∈G²₄(E) and points x∈ E\{q}are there such thata^lE(q) = (0,2,4) andl_E(−p−x)∈G¹₂(E)? The first condition restricts our attention to the linear series l_E = (O(4q), V) where V is a tridimensional vector space andH⁰(O(4q−2q))⊂V, while the second condition tells usH⁰(O(4q−p− problem discussed in [Far09b, Proof of Thm. 4.6]. The answer is

(g−1)15Ng−1,2,d,(0,2,2)+ 3Ng−1,2,d,(1,1,2)+ 3Ng−1,2,d,(0,1,3)

= 24(2s²+ 3s−4) s+ 3 N_g,2,d. Now let us suppose x ∈ C\ {y}. The condition on x and p can be reformulated in the following manner. We consider the curveC∪_yE as the special fiberX₀ of a family of curves π:X → B with sections x(t) and p(t) such that x(0) = x, p(0) = p, and with smooth general fiber having l = (L, V) a g²_d such that l(−x−p) is a g¹_d−2. Let V^′ ⊂V be the two dimensional linear subspace formed by those sectionsσ ∈V such that div(σ) ≥x+p. Then V^′ specializes onX₀ to V_C^′ ⊂V_C and V_E^′ ⊂V_E two-dimensional

3.2 The divisor D²_d that the map

ϕ:H⁰(lC)→H⁰(lC|2y+x)

has rank ≤1. By Thm. 3.0.2 there is only a finite number of such triples, and clearly the case a^lC(y)>(0,2,3) cannot occur. Moreover, note that xand y will be necessarily distinct.

Let µ=π_1,2,4:C×C×C×W_d²(C) →C×C×W_d²(C) andν =π_3,4:C×C×C× W_d²(C) → C×W_d²(C) be the natural projections respectively on the first, second and fourth components, and on the third and fourth components. Let π:C×C×W_d²(C)→ W_d²(C) be the natural projection on the third component. Now ϕ globalizes to

e

Let us find the Chern classes ofM. First we develop some notation (see also [ACGH85,

§VIII.2]). Letπ_i:C×C×C×W_d²(C)→C fori= 1,2,3 andπ₄:C×C×C×W_d²(C)→

Moreover

γ_ijγ_jk = η_jγ_ik, for 1≤i < j < k≤4. With this notation, we have

ch(ν^∗L ⊗O/I_D) = (1 +dη3+γ34−η3θ)1−e^{−(η1+γ13+η3+2η2+2γ23+2η3)}, hence by Grothendieck-Riemann-Roch

ch(M) =µ_∗((1 + (2−g)η3)ch(ν^∗L ⊗O/I_D))

= 3 + (d−2)η₁+ (2g+ 2d−6)η₂−2γ₁₂+γ₁₄+ 2γ₂₄

−η1θ−2η2θ+ (8−2d−4g)η1η2−2η1γ24−2η2γ14+ 2η1η2θ.

Using Newton’s identities, we recover the Chern classes ofM:

c₁(M) = (d−2)η1+ (2g+ 2d−6)η2−2γ12+γ₁₄+ 2γ24, c2(M) = (2d²−8d+ 2gd+ 8−4g)η1η2+ (2g+ 2d−8)η2γ14

+ (2d−4)η1γ₂₄+ 2γ14γ₂₄−2η2θ, c₃(M) = (4−2d)η1η₂θ−2η2γ₁₄θ.

We finally find

[Y] =η₁η₂(c²₁(2d²−8d+ 2dg+ 4−4(g−1))

+c₁θ(−12d−4g+ 40) +c₂(−4d+ 16−8g) + 12θ²)

= (28s+ 48)(s−2)(s−1)

(s+ 3) N_g,2,d·η₁η₂θ^g−1

where we have used the following identities proved in [Far09b, Lemma 2.6]

c²₁=

1 + 2s+ 2 s+ 3

c₂ c₁θ= (s+ 1)c2

θ²= (s+ 1)(s+ 2)

3 c₂

c₂=N_g,2,d·θ^g−1.

We are going to show that we have already considered all non zero contributions.

Indeed let us suppose x = y. Blowing up the point x, we obtain C ∪y P¹ ∪qE with x∈ P¹\ {y, q} and p ∈E\ {q}. We reformulate the condition on x and p viewing our curve as the special fiber of a family of curves π:X →B as before. Let{l_C, l_P¹, l_E} be a limitg²_d. Now V^′ specializes toV_C^′,V_P^′1 and V_E^′. There are three possibilities: either ρ(C, y) =ρ(P¹, x, y, q) =ρ(E, p, q) = 0, orρ(C, y) =−1,ρ(P¹, x, y, q) = 0,ρ(E, p, q) = 1, orρ(C, y) =−1,ρ(P¹, x, y, q) = 1,ρ(E, p, q) = 0. In all these casesa^lC(y) = (0,2, a^l₂^C(y)) (remember that lC is base-point free) and a^lE(q) = (a^l₀^E(q), d−2, d). Hence a^lP1(y) =

3.2 The divisor D²_d (a^l₀^P1(y), d−2, d) anda^lP1(q) = (0,2, a^l₂^P1(q)). Let us restrict now to the sections in V_C^′, V_P^′1 and V_E^′. For all sections σ_P¹ ∈V_P^′1 since div(σ_P¹)≥x, we have that ordy(σ_P¹) < d and hence ordy(σP¹) ≤ d−2. On the other side, since for all σ_E ∈ V_E^′, div(σE) ≥ p, we have that ordq(σE) < d and hence ordq(σP¹) ≥ 2. Let us take one section τ ∈V_P^′1

such that ordy(τ) = d−2. Since div(τ) ≥ (d−2)y +x, we get ordq(τ) ≤ 1, hence a contradiction.

Thus we have that

D²_d·C₁= 24(2s²+ 3s−4)

s+ 3 N_g,2,d+ (28s+ 48)(s−2)(s−1) (s+ 3) N_g,2,d.

while considering the intersection of the test curve C₁ with the generating classes we have

D²_d·C₁ =b₁(2g−4), whence

b₁= 14s³+ 6s²−8s (3s−2)(s+ 3)N_g,2,d.

Remark 3.2.2. The previous class [Y] being nonzero, it implies together with Thm.

3.0.2 that the scheme G_d²((0,2,3)) over M_g−1,1 is not irreducible.

3.2.4 The coefficient b_g−1

We analyze now the following test curve E. Let (C, p) be a general pointed curve of genusg−1 and (E, q) be a pointed elliptic curve. Let us identify the pointspandq and let y be a movable point inE. We have

0 =D²_d·E=c+b₁−b_g−1, whence

b_g−1 = 48s⁴+ 12s³−56s²+ 20s (3s−1)(3s−2)(s+ 3) N_g,2,d. 3.2.5 A test

Furthermore, as a test we consider the family of curves R. Let (C, p, q) be a general two-pointed curve of genusg−1 and let us identify the pointq with the base point of a general pencil of plane cubic curves. We have

0 =D²_d·R=a−12b0+b_g−1. 3.2.6 The remaining coefficients in case g = 6

Denote by P_g the moduli space of stable g-pointed rational curves. Let (E, p, q) be a general two-pointed elliptic curve and letj:Pg → Mg,1 be the map obtained identifying

the first marked point on a rational curve with the point q ∈ E and attaching a fixed elliptic tail at the other marked points. We claim thatj^∗(D²₆) = 0.

Indeed consider a flag curve of genus 6 in the image of j. Clearly the only possibility for the adjusted Brill-Noether numbers is to be zero on each aspect. In particular the collection of the aspects on all components but E smooths to a g²₆ on a general one-pointed curve of genus 5. As discussed in section 3.2.3, the point x can not be in E.

Supposexis in the rest of the curve. Then smoothing we getlag²₆ on a general pointed curve of genus 5 such thatl(−2q−x)) is ag¹₃, a contradiction.

Now let us study the pull-back of the generating classes. As in [EH87, §3] we have thatj^∗(λ) =j^∗(δ0) = 0. Furthermorej^∗(ψ) = 0.

For i = 1, . . . , g −3 denote by ε⁽¹⁾_i the class of the divisor which is the closure in P_g of the locus of two-component curves having exactly the first marked point and other i marked points on one of the two components. Then clearly j^∗(δi) = ε⁽¹⁾_i−1 for i= 2, . . . , g−2. Moreover adapting the argument in [EH89, pg. 49], we have that

j^∗(δg−1) =−

g−3X

i=1

i(g−i−1) g−2 ε⁽¹⁾_i while

j^∗(δ₁) =−

g−3X

i=1

(g−i−1)(g−i−2) (g−1)(g−2) ε⁽¹⁾_i . Finally sincej^∗(D²₆) = 0, checking the coefficient of ε⁽¹⁾_i we obtain

b_i+1 = (g−i−1)(g−i−2)

(g−1)(g−2) b₁+i(g−i−1) g−2 b_g−1 fori= 1,2,3.

Bibliography

[AC81a] Arbarello, Enrico; Cornalba, Maurizio: Footnotes to a paper of Beniamino Segre: “On the modules of polygonal curves and on a complement to the Rie-mann existence theorem” (Italian) [Math. Ann. 100 (1928), 537–551; Jbuch 54, 685]. In: Math. Ann., volume 256(3):pp. 341–362, 1981. ISSN 0025-5831.

doi:10.1007/BF01679702. The number of g¹_d’s on a general d-gonal curve, and the unirationality of the Hurwitz spaces of 4-gonal and 5-gonal curves, URLhttp://dx.doi.org/10.1007/BF01679702.

[AC81b] Arbarello, Enrico; Cornalba, Maurizio: On a conjecture of Petri. In: Com-ment. Math. Helv., volume 56(1):pp. 1–38, 1981. ISSN 0010-2571. doi:

10.1007/BF02566195. URLhttp://dx.doi.org/10.1007/BF02566195.

[AC87] Arbarello, Enrico; Cornalba, Maurizio: The Picard groups of the moduli spaces of curves. In: Topology, volume 26(2):pp. 153–171, 1987. ISSN 0040-9383. doi:10.1016/0040-9383(87)90056-5. URL http://dx.doi.org/10.1016/0040-9383(87)90056-5.

[AC96] Arbarello, Enrico; Cornalba, Maurizio: Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. In: J. Algebraic Geom., volume 5(4):pp. 705–749, 1996. ISSN 1056-3911.

[ACGH85] Arbarello, E.; Cornalba, M.; Griffiths, P. A.; Harris, J.: Geometry of alge-braic curves. Vol. I, volume 267 of Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1985. ISBN 0-387-90997-4.

[BN74] Brill, Alexander von; Nöether, Max: Über die algebraischen Functionen und ihre Anwendung in der Geometrie. In: Clebsch Ann., volume VII:pp. 269–

316, 1874.

[Cas89] Castelnuovo, Guido: Numero delle involuzioni razionali giacenti sopra una curva di dato genere. In: Rendiconti R. Accad. Lincei, volume 5(4):pp. 130–

133, 1889.

[Cuk89] Cukierman, Fernando: Families of Weierstrass points. In:

Duke Math. J., volume 58(2):pp. 317–346, 1989. ISSN 0012-7094. doi:10.1215/S0012-7094-89-05815-8. URL http://dx.doi.org/10.1215/S0012-7094-89-05815-8.

[Deb01] Debarre, Olivier: Higher-dimensional algebraic geometry. Universitext.

Springer-Verlag, New York, 2001. ISBN 0-387-95227-6.

[Dia85] Diaz, Steven: Exceptional Weierstrass points and the divisor on moduli space that they define. In: Mem. Amer. Math. Soc., volume 56(327):pp. iv+69, 1985. ISSN 0065-9266.

[Edi92] Edidin, Dan: The codimension-two homology of the moduli space of stable curves is algebraic. In: Duke Math. J., volume 67(2):pp. 241–

272, 1992. ISSN 0012-7094. doi:10.1215/S0012-7094-92-06709-3. URL http://dx.doi.org/10.1215/S0012-7094-92-06709-3.

[Edi93] Edidin, Dan: Brill-Noether theory in codimension-two. In: J. Algebraic Geom., volume 2(1):pp. 25–67, 1993. ISSN 1056-3911.

[EH83a] Eisenbud, David; Harris, Joe: Divisors on general curves and cus-pidal rational curves. In: Invent. Math., volume 74(3):pp. 371–

418, 1983. ISSN 0020-9910. doi:10.1007/BF01394242. URL http://dx.doi.org/10.1007/BF01394242.

[EH83b] Eisenbud, David; Harris, Joe: A simpler proof of the Gieseker-Petri theorem on special divisors. In: Invent. Math., volume 74(2):pp.

269–280, 1983. ISSN 0020-9910. doi:10.1007/BF01394316. URL http://dx.doi.org/10.1007/BF01394316.

[EH86] Eisenbud, David; Harris, Joe: Limit linear series: basic theory. In:

Invent. Math., volume 85(2):pp. 337–371, 1986. ISSN 0020-9910. doi:

10.1007/BF01389094. URLhttp://dx.doi.org/10.1007/BF01389094.

[EH87] Eisenbud, David; Harris, Joe: The Kodaira dimension of the moduli space of curves of genus ≥ 23. In: Invent. Math., volume 90(2):pp.

359–387, 1987. ISSN 0020-9910. doi:10.1007/BF01388710. URL http://dx.doi.org/10.1007/BF01388710.

[EH89] Eisenbud, David; Harris, Joe: Irreducibility of some families of lin-ear series with Brill-Noether number −1. In: Ann. Sci. École Norm.

Sup. (4), volume 22(1):pp. 33–53, 1989. ISSN 0012-9593. URL http://www.numdam.org/item?id=ASENS_1989_4_22_1_33_0.

[Fab88] Faber, Carel: Chow rings of moduli spaces of curves. Ph.D. thesis, Amster-dam, 1988.

[Fab89] Faber, Carel: Some results on the codimension-two Chow group of the moduli space of stable curves. In: Algebraic curves and projec-tive geometry (Trento, 1988), volume 1389 of Lecture Notes in Math., pp. 66–75. Springer, Berlin, 1989. doi:10.1007/BFb0085924. URL http://dx.doi.org/10.1007/BFb0085924.

Bibliography [Fab90a] Faber, Carel: Chow rings of moduli spaces of curves. I. The Chow ring ofM3. In: Ann. of Math. (2), volume 132(2):pp. 331–419, 1990. ISSN 0003-486X.

doi:10.2307/1971525. URLhttp://dx.doi.org/10.2307/1971525.

[Fab90b] Faber, Carel: Chow rings of moduli spaces of curves. II. Some re-sults on the Chow ring of M4. In: Ann. of Math. (2), volume 132(3):pp. 421–449, 1990. ISSN 0003-486X. doi:10.2307/1971526. URL http://dx.doi.org/10.2307/1971526.

[Fab99] Faber, Carel: A conjectural description of the tautological ring of the moduli space of curves. In: Moduli of curves and abelian varieties, Aspects Math., E33, pp. 109–129. Vieweg, Braunschweig, 1999.

[Far00] Farkas, Gavril: The geometry of the moduli space of curves of genus 23. In:

Math. Ann., volume 318(1):pp. 43–65, 2000. ISSN 0025-5831. doi:10.1007/

s002080000108. URLhttp://dx.doi.org/10.1007/s002080000108.

[Far08] Farkas, Gavril: Higher ramification and varieties of secant divisors on the generic curve. In: J. Lond. Math. Soc. (2), volume 78(2):pp.

418–440, 2008. ISSN 0024-6107. doi:10.1112/jlms/jdn038. URL http://dx.doi.org/10.1112/jlms/jdn038.

[Far09a] Farkas, Gavril: Birational aspects of the geometry of Mg. In: Surveys in differential geometry. Vol. XIV. Geometry of Riemann surfaces and their moduli spaces, volume 14 of Surv. Differ. Geom., pp. 57–110. Int. Press, Somerville, MA, 2009.

[Far09b] Farkas, Gavril: Koszul divisors on moduli spaces of curves. In: Amer. J.

Math., volume 131(3):pp. 819–867, 2009. ISSN 0002-9327. doi:10.1353/ajm.

0.0053. URLhttp://dx.doi.org/10.1353/ajm.0.0053.

[Far10] Farkas, Gavril: Rational maps between moduli spaces of curves and Gieseker-Petri divisors. In: J. Algebraic Geom., volume 19(2):pp. 243–284, 2010. ISSN 1056-3911.

[FP05] Faber, Carel; Pandharipande, Rahul: Relative maps and tautological classes.

In: J. Eur. Math. Soc. (JEMS), volume 7(1):pp. 13–49, 2005. ISSN 1435-9855. doi:10.4171/JEMS/20. URLhttp://dx.doi.org/10.4171/JEMS/20.

[Ful69] Fulton, William: Hurwitz schemes and irreducibility of moduli of algebraic curves. In: Ann. of Math. (2), volume 90:pp. 542–575, 1969. ISSN 0003-486X.

[Ful98] Fulton, William: Intersection theory, volume 2 ofErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2nd edition, 1998. ISBN 3-540-62046-X; 0-387-98549-2.

[FV09] Farkas, Gavril; Verra, Alessandro: The intermediate type of certain moduli spaces of curves. In: Preprint, arXiv:0910.3905, 2009.

[GH80] Griffiths, Phillip; Harris, Joe: On the variety of special lin-ear systems on a general algebraic curve. In: Duke Math.

J., volume 47(1):pp. 233–272, 1980. ISSN 0012-7094. URL http://projecteuclid.org/getRecord?id=euclid.dmj/1077313873.

[Har84] Harris, Joe: On the Kodaira dimension of the moduli space of curves. II. The even-genus case. In: Invent. Math., volume 75(3):pp.

437–466, 1984. ISSN 0020-9910. doi:10.1007/BF01388638. URL http://dx.doi.org/10.1007/BF01388638.

[HM82] Harris, Joe; Mumford, David: On the Kodaira dimension of the moduli space of curves. In: Invent. Math., volume 67(1):pp. 23–88, 1982. ISSN 0020-9910. doi:10.1007/BF01393371. With an appendix by William Fulton, URLhttp://dx.doi.org/10.1007/BF01393371.

[HM90] Harris, Joe; Morrison, Ian: Slopes of effective divisors on the mod-uli space of stable curves. In: Invent. Math., volume 99(2):pp.

321–355, 1990. ISSN 0020-9910. doi:10.1007/BF01234422. URL http://dx.doi.org/10.1007/BF01234422.

[HM98] Harris, Joe; Morrison, Ian: Moduli of curves, volume 187 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1998. ISBN 0-387-98438-0;

0-387-98429-1.

[Jen10] Jensen, David: Rational fibrations of M5,1 and M6,1. In: Preprint, arXiv:1012.5115, 2010.

[Kho07] Khosla, Deepak: Tautological classes on moduli spaces of curves with linear series and a push-forward formula when ρ = 0. In: Preprint, arXiv:0704.1340, 2007.

[Laz86] Lazarsfeld, Robert: Brill-Noether-Petri without degenerations. In: J. Dif-ferential Geom., volume 23(3):pp. 299–307, 1986. ISSN 0022-040X. URL http://projecteuclid.org/getRecord?id=euclid.jdg/1214440116.

[Log03] Logan, Adam: The Kodaira dimension of moduli spaces of curves with marked points. In: Amer. J. Math., volume 125(1):pp. 105–138, 2003. ISSN 0002-9327.

[Mar46] Maroni, Arturo: Le serie lineari speciali sulle curve trigonali. In: Ann. Mat.

Pura Appl. (4), volume 25:pp. 343–354, 1946. ISSN 0003-4622.

[MS86] Martens, Gerriet; Schreyer, Frank-Olaf: Line bundles and syzygies of trigonal curves. In: Abh. Math. Sem. Univ. Hamburg, volume 56:pp. 169–189, 1986.

ISSN 0025-5858.

Bibliography [Mum77] Mumford, David: Stability of projective varieties. In: Enseignement Math.

(2), volume 23(1-2):pp. 39–110, 1977. ISSN 0013-8584.

[Mum83] Mumford, David: Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and geometry, Vol. II, volume 36 ofProgr. Math., pp.

271–328. Birkhäuser Boston, Boston, MA, 1983.

[SB89] Shepherd-Barron, Nicholas I.: Invariant theory forS₅ and the rationality of M₆. In: Compositio Math., volume 70(1):pp. 13–25, 1989. ISSN 0010-437X.

URLhttp://www.numdam.org/item?id=CM_1989__70_1_13_0.

[SF00] Stankova-Frenkel, Zvezdelina E.: Moduli of trigonal curves. In: J. Algebraic Geom., volume 9(4):pp. 607–662, 2000. ISSN 1056-3911.

[Ste98] Steffen, Frauke: A generalized principal ideal theorem with an appli-cation to Brill-Noether theory. In: Invent. Math., volume 132(1):pp.

73–89, 1998. ISSN 0020-9910. doi:10.1007/s002220050218. URL http://dx.doi.org/10.1007/s002220050218.

Im Dokument Geometric cycles on moduli spaces of curves (Seite 74-89)