**RATIONAL MAPS BETWEEN MODULI SPACES OF CURVES AND**
**GIESEKER-PETRI DIVISORS**

GAVRIL FARKAS

For a a general smooth projective curve[C] ∈ Mg and an arbitrary line bundle
L∈Pic(C), the*Gieseker-Petri*theorem states that the multiplication map

µ_{0}(L) :H^{0}(C, L)⊗H^{0}(C, K_{C}⊗L^{∨})→H^{0}(C, K_{C})

is injective. The theorem, conjectured by Petri and proved by Gieseker [G] (see [EH3]

for a much simplified proof), lies at the cornerstone of the theory of algebraic curves. It
implies that the varietyG^{r}_{d}(C) = {(L, V) :L ∈Pic^{d}(C), V ∈G(r+ 1, H^{0}(L))}of linear
series of degreedand dimensionr is smooth and of expected dimensionρ(g, r, d) :=

g−(r+ 1)(g−d+r)and that the forgetful mapG^{r}_{d}(C)→W_{d}^{r}(C)is a rational resolution
of singularities (see [ACGH] for many other applications). It is an old open problem
to describe the locusGPg ⊂ Mg consisting of curves[C]∈ Mgsuch that there exists a
line bundleLonC for which the Gieseker-Petri theorem fails. ObviouslyGPg breaks
up into irreducible components depending on the numerical types of linear series. For
fixed integersd, r ≥ 1 such that g−d+r ≥ 2, we define the locusGP^{r}_{g,d} consisting
of curves [C] ∈ Mg such that there exist a pair of linear series (L, V) ∈ G^{r}_{d}(C) and
(K_{C} ⊗L^{∨}, W)∈G^{g−d+r−1}_{2g−2−d} (C)for which the multiplication map

µ_{0}(V, W) :V ⊗W →H^{0}(C, K_{C})

is not injective. Even though certain components ofGPgare well-understood, its global
geometry seems exceedingly complicated. Ifρ(g, r, d) =−1, thenGP^{r}_{g,d}coincides with
the Brill-Noether divisor M^{r}_{g,d} of curves[C] ∈ M_{g} with G^{r}_{d}(C) 6= ∅ which has been
studied by Eisenbud and Harris in [EH2] and used to prove thatMg is of general type
forg ≥ 24. The locusGP^{1}_{g,g−1} can be identified with the divisor of curves carrying a
vanishing theta-null and this has been studied by Teixidor (cf. [T]). We proved in [F2]

that forr = 1and(g+ 2)/2 ≤ d ≤ g−1, the locusGP^{1}_{g,d} always carries a divisorial
component. It is conjectured that the locusGPgis pure of codimension1inMgand we
go some way towards proving this conjecture. Precisely, we show thatGPgis supported
in codimension1for every possible numerical type of a linear series:

**Theorem 0.1.** *For any positive integers*g, d*and*r*such that*ρ(g, r, d)≥0*and*g−d+r ≥2,
*the locus*GP^{r}_{g,d}*has a divisorial component in*Mg*.*

The main issue we address in this paper is a detailed intersection theoretic study of a rational map between two different moduli spaces of curves. We fixg:= 2s+ 1≥3.

Research partially supported by an Alfred P. Sloan Fellowship, the NSF Grants DMS-0450670 and DMS- 0500747 and a 2006 Texas Summer Research Assignment. Most of this paper has been written while visiting the Institut Mittag-Leffler in Djursholm in the Spring of 2007. Support from the institute is gratefully acknowledged.

1

Sinceρ(2s+1,1, s+2) = 1we can define a rational map between moduli spaces of curves
φ:M_{2s+1}− −>M_{1+} ^{s}

s+1(^{2s+2}_{s} ), φ([C]) := [W_{s+2}^{1} (C)].

The fact thatφis well-defined, as well as a justification for the formula of the genusg^{′} :=

g(W_{s+2}^{1} (C))of the curve of special divisors of typeg^{1}_{s+2}, is given in Section 3. It is known
thatφis generically injective (cf. [PT], [CHT]). Sinceφis the only-known rational map
between two moduli spaces of curves and one of the very few natural examples of a
rational map admitted byMg, its study is clearly of independent interest. In this paper
we carry out a detailed enumerative study ofφand among other things, we determine
the pull-back mapφ^{∗} :Pic(Mg^{′})→ Pic(Mg)(see Theorem 3.4 for a precise statement).

In particular we have the following formula concerning slopes of divisor classes pulled
back fromMg^{′} (For the definition of the slope functions: Eff(Mg) →R∪ {∞}on the
cone of effective divisors we refer to [HMo] or [FP]):

**Theorem 0.2.** *We set*g:= 2s+1*and*g^{′} := 1+_{s+1}^{s} ^{2s+2}_{s}

*. For any divisor class*D∈Pic(M_{g}^{′})
*having slope*s(D) =c, we have the following formula for the slope ofφ^{∗}(D)∈Pic(Mg):

s(φ^{∗}(D)) = 6 + 8s^{3}(c−4) + 5cs^{2}−30s^{2}+ 20s−8cs−2c+ 24
s(s+ 2)(cs^{2}−4s^{2}−c−s+ 6) .

We use this formula to describe the cone Mov(Mg)of*moving divisors*^{1}inside the
cone Eff(Mg) of effective divisors. The cone Mov(Mg) parameterizes rational maps
fromMgin the projective category while the coneNef(Mg)of numerically effective di-
visors, parameterizes regular maps fromM_{g}(see [HK] for details on this perspective).

A fundamental question is to estimate the following slope invariants associated toMg:
s(M_{g}) := inf_{D∈Eff(M}_{g}_{)}s(D) and s^{′}(M_{g}) := inf_{D∈Mov(M}_{g}_{)}s(D).

The formula of the class of Brill-Noether divisorsM^{r}_{g,d}whenρ(g, r, d) =−1shows that
lim_{g→∞}s(Mg)≤6(cf. [EH2]). In [F1] we provided an infinite sequence of genera of the
formg=a(2a+ 1)witha≥2for whichs(Mg)<6 + 12/(g+ 1), thus contradicting the
Slope Conjecture [HMo]. There is no known example of a genusgsuch thats(M_{g})<6.

Understanding the difference between s(Mg) and s^{′}(Mg) is a subtle question
even for lowg. There is a strict inequalitys(Mg) < s^{′}(Mg)whenever one can find an
effective divisorD ∈ Eff(Mg) with s(D) = s(Mg), such that there exists a covering
curve R ⊂ D for which R ·D < 0. For g < 12 the divisors minimizing the slope
function have a strong geometric characterization in terms of Brill-Noether theory. Thus
computings^{′}(Mg)becomes a problem in understanding the geometry of Brill-Noether
and Gieseker-Petri divisors onMg. To illustrate this point we give two examples (see
Section 5 for details): It is known thats(M_{3}) = 9and the minimum slope is realized by
the locus of hyperelliptic curvesM^{1}_{3,2} ≡ 9λ−δ_{0}−3δ_{1}. However[M^{1}_{3,2}] ∈/ Mov(M3),
because M^{1}_{3,2} is swept out by pencils R ⊂ M3 with R·δ/R ·λ = 28/3 > s(M^{1}_{3,2}).

1Recall that an effectiveQ-Cartier divisorDon a normal projective varietyXis said to be moving, if the stable base locusT

n≥1Bs|OX(nD)|has codimension at least2inX.

In fact, one has equalitys^{′}(M3) = 28/3and the moving divisor onM3 attaining this
bound corresponds to the pull-back of an ample class under the rational map

M_{3}− −>Q_{4} :=|O_{P}^{2}(4)|//SL(3)

to the GIT quotient of plane quartics which contracts M^{1}_{3,2} to a point (see [HL] for
details on the role of this map in carrying out the Minimal Model Program forM_{3}).

Forg= 10, it is known thats(M10) = 7and this bound is attained by the divisor
K_{10}of curves lying onK3surfaces (cf. [FP] Theorems 1.6 and 1.7 for details):

K10≡7λ−δ_{0}−5δ_{1}−9δ_{2}−12δ_{3}−14δ_{4}−15δ_{5}.

Furthermore,K10is swept out by pencilsR ⊂ M10withR·δ/R·λ= 78/11 > s(K10)
(cf. [FP] Proposition 2.2). Therefore[K10]∈/ Mov(M10)ands^{′}(M10)≥78/11.

Forg= 2s+1using the elementary observation thatφ^{∗}(Ample(Mg^{′}))⊂Mov(Mg),
Theorem 0.2 provides a uniform upper bound on slopes of moving divisors onMg:
**Corollary 0.3.** *We set*g:= 2s+ 1*as and*g^{′} := 1 + _{s+1}^{s} ^{2s+2}_{s}

*as above. Then*
s(φ^{∗}(D))<6 + 16

g−1 *for every divisor*D∈Ample(M_{g}^{′}).

*In particular one has the estimate*s^{′}(Mg)<6 + 16/(g−1)*, for every odd integer*g≥3.

Since we also know thatlimg→∞s(Mg) ≤6, Corollary 0.3 indicates that (at least
asymptotically, for largeg) we cannot distinguish between effective and moving divi-
sors onMg. We ask whether it is true thatlim_{g→∞}s(Mg) = lim_{g→∞}s^{′}(Mg)?

At the heart of the description in codimension1of the map φ: Mg− − >Mg^{′}

lies the computation of the cohomology class of the compactified Gieseker-Petri divisor
GP^{r}_{g,d} ⊂ Mg in the case when ρ(g, r, d) = 1. Since this calculation is of independent
interest we discuss it in some detail. We denote byG^{r}_{d} the stack parameterizing pairs
[C, l]with[C]∈ Mg andl = (L, V) ∈ G^{r}_{d}(C)and denote byσ :G^{r}_{d} → Mg the natural
projection. In [F1] we computed the class of GP^{r}_{g,d} in the case ρ(g, r, d) = 0, when
GP^{r}_{g,d} can be realized as the push-forward of a determinantal divisor onG^{r}_{d}under the
generically finite map σ. In particular, we showed that if we write g = rs+s and
d = rs+r wherer ≥ 1ands ≥ 2(hence ρ(g, r, d) = 0), then we have the following
formula for the slope ofGP^{r}_{g,d}(cf. [F1], Theorem 1.6):

s(GP^{r}_{g,d}) = 6 + 12

g+ 1+ 6(s+r+ 1)(rs+s−2)(rs+s−1) s(s+ 1)(r+ 1)(r+ 2)(rs+s+ 4)(rs+s+ 1).

The number 6 + 12/(g + 1) is the slope of all Brill-Noether divisors on Mg, that is
s(GP^{r}_{g,d}) = 6 + 12/(g+ 1)wheneverρ(g, r, d) =−1(cf. [EH2], or [F1] Corollary 1.2 for a
different proof, making use of M. Green’s Conjecture on syzygies of canonical curves).

In the technically much-more intricate case ρ(g, r, d) = 1, we can realize GP^{r}_{g,d}
as the push-forward of a codimension 2determinantal subvariety of G^{r}_{d} and most of
Section 2 is devoted to extending this structure over a partial compactification of Mg

corresponding to tree-like curves. Ifσ:Ge^{r}_{d}→Mfgdenotes the stack of limit linear series

g^{r}_{d}, we construct two*locally free*sheavesF andN overGe^{r}_{d}such that rank(F) = r+ 1,
rank(N) =g−d+r=:srespectively, together with a vector bundle morphism

µ:F ⊗ N →σ^{∗} E⊗ O_{M}_{g}(

[g/2]X

j=1

(2j−1)·δj)
such thatGP^{r}_{g,d}is the push-forward of the first degeneration locus ofµ:

**Theorem 0.4.** *We fix integers*r, s ≥ 1*and we set* g := rs+s+ 1, d := rs+r + 1*so that*
ρ(g, r, d) = 1. Then the class of the compactified Gieseker-Petri divisorGP^{r}_{g,d}*in*M_{g} *is given*
*by the formula:*

GP^{r}_{g,d}≡ C_{r+1}(s−1)r

2(r+s+ 1)(s+r)(r+s+ 2)(rs+s−1) aλ−b_{0}δ_{0}−b_{1}δ_{1}−

[g/2]X

j=2

b_{j}δ_{j}
,
*where*

C_{r+1}:= (rs+s)!r! (r−1)!· · ·2! 1!

(s+r)! (s+r−1)!· · ·(s+ 1)!s!

a= 2s^{3}(s+ 1)r^{5}+s^{2}(2s^{3}+ 14s^{2}+ 33s+ 25)r^{4}+s(10s^{4}+ 59s^{3}+ 162s^{2}+ 179s+ 54)r^{3}+
+(18s^{5}+138s^{4}+387s^{3}+491s^{2}+244s+24)r^{2}+(14s^{5}+145s^{4}+464s^{3}+627s^{2}+378s+72)r+

4s^{5}+ 54s^{4}+ 208s^{3}+ 314s^{2}+ 212s+ 48

b_{0} := (r+ 2)(s+ 1)(s+r+ 1)(2rs+ 2s+ 1)(rs+s+ 2)(rs+s+ 6)
6

b_{1} := (r+ 1)s

2s^{2}(s+ 1)r^{4}+s(2s^{3}+ 12s^{2}+ 23s+ 9s)r^{3}+ (8s^{4}+ 39s^{3}+ 75s^{2}+ 46s+ 10)r^{2}+
+(10s^{4}+ 59s^{3}+ 108s^{2}+ 89s+ 26)r+ 4s^{4}+ 30s^{3}+ 64s^{2}+ 58s+ 12

,
*and*b_{j} ≥b_{1}*for*j≥2*are explicitly determined constants.*

Even though the coefficientsaandb_{1}look rather unwieldy, the expression for the
slope ofGP^{r}_{g,d}has a simpler and much more suggestive expression which we record:

**Corollary 0.5.** *For*ρ(g, r, d) = 1*, the slope of the Gieseker-Petri divisor*GP^{r}_{g,d}*has the following*
*expression:*

s(GP^{r}_{g,d}) = 6 + 12

g+ 1+ 24s(r+ 1)(r+s)(s+r+ 2)(rs+s−1)

(r+ 2)(s+ 1)(s+r+ 1)(2rs+ 2s+ 1)(rs+s+ 2)(rs+s+ 6).
Next we specialize to the caser = 1, thusg = 2s+ 1. Using the base point free
pencil trick one can see that the divisorGP^{1}_{2s+1,s+2} splits into two irreducible compo-
nents according to whether the pencil for which the Gieseker-Petri theorem fails has a
base point or not. Precisely we have the following equality of codimension1cycles

GP^{1}_{2s+1,s+2}= (2s−2)· M^{1}_{2s+1,s+1}+GP^{1,0}_{2s+1,s+2},

whereGP^{1,0}_{2s+1,s+2} is the closure of the locus of curves[C] ∈ Mg carrying a base point
free pencilL ∈ W_{s+2}^{1} (C)such thatµ_{0}(L)is not injective. Since we also have the well-
known formula for the class of the Hurwitz divisor (cf. [EH2], Theorem 1)

M^{1}_{2s+1,s+1}≡ (2s−2)!

(s+ 1)! (s−1)!

6(s+ 2)λ−(s+ 1)δ_{0}−6sδ_{1}− · · ·
,

we find the following expression for the slope ofGP^{1,0}_{2s+1,s+2}:

**Corollary 0.6.** *For*g = 2s+ 1, the slope of the divisorGP^{1,0}_{2s+1,s+2} *of curves carrying a base*
*point free pencil*L∈W_{s+2}^{1} (C)*such that*µ0(L)*is not injective, is given by the formula*

s(GP^{1,0}_{2s+1,s+2}) = 6 + 12

g+ 1+ 2s−1 (s+ 1)(s+ 2).

We note that fors= 2andg= 5, the divisorGP^{1,0}_{5,4}is equal to Teixidor’s divisor of
curves[C]∈ M5 having a vanishing theta-null, that is, a theta-characteristicL^{⊗2}=KC

withh^{0}(C, L) ≥2. In this case Corollary 0.6 specializes to her formula [T] Theorem 3.1:

GP^{1,0}_{5,4} ≡4·(33λ−4δ_{0}−15δ_{1}−21δ_{2})∈Pic(M_{5}).

To give another example we specialize to the caser= 1,s= 3wheng= 7. Using
the base point free pencil trick, the divisorGP^{1}_{7,5} can be identified with the closure of
the locus of curves[C] ∈ M_{7} possessing a linear series l ∈ G^{2}_{7}(C) such that the plane
modelC→^{l} **P**^{2} has8nodes, of which7lie on a conic. Its class is given by the formula:

GP^{1}_{7,5}≡4·(201λ−26δ0−111δ1−177δ2−198δ3)∈Pic(M7).

In Section 5 we shall need a characterization of thek-gonal lociM^{1}_{g,k} in terms of
effective divisors ofMgcontaining them. For instance, it is known that ifD∈Eff(Mg)
is a divisor such thats(D)<8 + 4/g, thenDcontains the hyperelliptic locusM^{1}_{g,2}(see
e.g. [HMo], Corollary 3.30). Similar bounds exist for the trigonal locus: ifs(D)<7+6/g
thenD⊃ M^{1}_{g,3}. We have the following extension of this type of result:

**Theorem 0.7.** *1) Every effective divisor*D ∈Eff(Mg) *having slope*s(D) < ^{1}_{g}_{13g+16}

2

*con-*
*tains the locus*M^{1}_{g,4}*of*4-gonal curves.

*2) Every effective divisor*D∈Eff(Mg)*having slope*s(D)< ^{1}_{g}

5g+ 9 + 2[^{g+1}_{2} ]

*contains the*
*locus*M^{1}_{g,5}*of*5-gonal curves.

The proof uses an explicit unirational parametrization ofM^{1}_{g,k} that is available
only whenk ≤5. It is natural to ask whether the subvarietyM^{1}_{g,k} ⊂ M_{g} is cut out by
divisors D ∈ Eff(Mg) of slope less than the bound given in Theorem 0.7. Very little
seems to be known about this question even in the hyperelliptic case.

We close by summarizing the structure of the paper. In Section 1 we introduce a
certain stack of pairs of complementary limit linear series which we then use to prove
Theorem 0.1 by induction on the genus. The class of the compactified Gieseker-Petri
divisor is computed in Section 2. This calculation is used in Section 3 to describe maps
between moduli spaces of curves. We then study the geometry ofφin low genus (Sec-
tion 4) with applications to Prym varieties and we finish the paper by computing the
invariants^{′}(Mg)forg≤11(Section 5).

1. DIVISORIAL COMPONENTS OF THEG^{IESEKER}-P^{ETRI LOCUS}

Let us fix positive integersg, randgsuch thatρ(g, r, d)≥0and sets:=g−d+r ≥
2, henceg=rs+s+jandd=rs+r+j, withj≥0. The casej= 0corresponds to the
situationρ(g, r, d) = 0when we already know thatGP^{r}_{g,d}has a divisorial component in
Mg whose class has been computed (see [F1], Theorem 1.6). We present an inductive
method on j which produces a divisorial component of GP^{r}_{g,d} ⊂ Mg provided one
knows that GP^{r}_{g−1,d−1} has a divisorial component in Mg−1. The method is based on
degeneration to the boundary divisor ∆_{1} ⊂ Mg and is somewhat similar to the one
used in [F2] for the caser = 1.

We briefly recall a few facts about (degeneration of) multiplication maps on curves.

IfLandM are line bundles on a smooth curveC, we denote by
µ_{0}(L, M) :H^{0}(L)⊗H^{0}(M)→H^{0}(L⊗M)
the usual multiplication map and by

µ1(L, M) :Kerµ0(L, M)→H^{0}(KC⊗L⊗M), µ1(X

i

σi⊗τi) :=X

i

(dσi)·τi,
the first Gaussian map associated toLandM (see [W]). For anyρ ∈ H^{0}(L)⊗H^{0}(M)
and a pointp∈C, we write that ordp(ρ)≥k, ifρlies in the span of elements of the form
σ⊗τ, where σ ∈ H^{0}(L)andτ ∈ H^{0}(M)are such that ordp(σ) +ordp(τ) ≥ k. When
i= 0,1, the condition ordp(ρ)≥i+ 1for a generic pointp ∈C, is clearly equivalent to
ρ∈Kerµ_{i}(L, M).

If X is a tree-like curve and l is a limit g^{r}_{d} onX, for an irreducible component
Y ⊂ X we denote by l_{Y} = (L_{Y}, V_{Y} ⊂ H^{0}(L_{Y})) the Y-aspect of l. For p ∈ Y we
denote by{a^{l}_{i}^{Y}(p)}i=0...rthe*vanishing sequence*oflatpand byρ(l_{Y}, p) :=ρ(g(Y), r, d)−
Pr

i=0(a^{l}_{i}^{Y}(p)−i)the*adjusted Brill-Noether number*with respect to the pointp(see [EH1]

for a general reference on limit linear series).

We shall repeatedly use the following elementary observation already made in
[EH3] and used in [F2]: Suppose{σi} ⊂H^{0}(L)and{τj} ⊂ H^{0}(M)are bases of global
sections with the property that ordp(σi) = a^{L}_{i}(p)and ordp(τj) = a^{M}_{j} (p)for alliandj.

Then ifρ ∈Kerµ_{0}(L, M)), there must exist two pairs of integers(i_{1}, j_{1}) 6= (i2, j_{2})such
that ordp(ρ) =ordp(σi1) +ordp(τj1) =ordp(σi2) +ordp(τj2).

A technical tool in the paper is the stackν:Ue_{g,d}^{r} →Mfgof pairs of complementary
limit linear series defined over a partial compactification ofM_{g} which will be defined
below. ThenGP^{r}_{g,d}is the push-forward underν_{|ν}^{−1}_{(M}_{g}_{)}of a degeneration locus inside
Ue_{g,d}^{r} . We denote byPic^{d}the degreedPicard stack overMg, that is, the ´etale sheafifica-
tion of the Picard functor, and byEthe Hodge bundle overMg. We considerG^{r}_{d}⊂Pic^{d}
to be the stack parameterizing pairs[C, l]with l = (L, V) ∈ G^{r}_{d}(C) and the projection
σ:G^{r}_{d}→ Mg.

We set∆^{0}_{0} ⊂∆_{0} ⊂ M_{g} to be the locus of curves[C/y∼q], where[C, q]∈ M_{g−1,1}
is Brill-Noether general andy ∈C is an arbitrary point, as well as their degenerations
[C∪qE∞], whereE∞is a rational nodal curve, that is,j(E∞) =∞. For1≤i≤[g/2], we

denote by∆^{0}_{i} ⊂∆i the open subset consisting of unions[C∪y D], where[C]∈ Miand
[D, y]∈ Mg−i,1are Brill-Noether general curves but the pointy ∈Cis arbitrary. Then
if we denote byMfg :=Mg∪ ∪^{[g/2]}_{i=0} ∆^{0}_{i}

, one can extend the coveringσ:G^{r}_{d}→ Mgto a
proper mapσ:Ge^{r}_{d}→Mfgfrom the stackGe^{r}_{d}of limit linear seriesg^{r}_{d}.

We now introduce the stackν : Ue_{g,d}^{r} → Mfg of complementary linear series: For
[C] ∈ Mg, the fibre ν^{−1}[C]parameterizes pairs(l, m) wherel = (L, V) ∈ G^{r}_{d}(C) and
m = (KC ⊗L^{∨}, W) ∈ G^{g−d+r−1}_{2g−2−d} (C). If[C = C1 ∪y C2] ∈ Mfg, where [C1, y] ∈ Mi,1

and[C_{2}, y]∈ Mg−i,1, the fibreν^{−1}[C]consists of pairs of limit linear series(l, m), where
l={(LC1, VC1),(LC2, VC2)}is a limitg^{r}_{d}onCand

m={ K_{C}1 ⊗ O_{C}1(2(g−i)·p)⊗L^{−1}_{C}_{1}, W_{C}1

, K_{C}2 ⊗ O_{C}2(2i·p)⊗L^{−1}_{C}_{2}, W_{C}2

}
is a limit g^{g−d+r−1}_{2g−2−d} onC which is complementary tol. There is a morphism of stacks
ǫ:Ue_{g,d}^{r} →Ge^{r}_{g,d}which forgets the limitg^{g−d+r−1}_{2g−2−d} on each curve. Clearlyσ◦ǫ=ν.

**Definition 1.1.** For a smooth curveCof genusg, a Gieseker-Petri(gp)^{r}_{d}*-relation*consists
of a pair of linear series(L, V) ∈ G^{r}_{d}(C)and(K_{C} ⊗L^{∨}, W) ∈ G^{g−d+r−1}_{2g−2−d} (C), together
with an elementρ∈**PKer{µ**0(V, W) :V ⊗W →H^{0}(K_{C})}.

IfC =C_{1}∪pC_{2}is a curve of compact type withC_{1} andC_{2}being smooth curves
with g(C_{1}) = i and g(C_{2}) = g−i respectively, a (gp)^{r}_{d}*-relation* on C is a collection
(l, m, ρ_{1}, ρ_{2}), where[C, l, m]∈Ue_{g,d}^{r} , and elements

ρ_{1} ∈**PKer{V**C1⊗W_{C}_{1} →H^{0} K_{C}_{1}(2(g−i)p)

}, ρ2∈**PKer{V**C2⊗W_{C}_{2} →H^{0} K_{C}_{2}(2ip)
}
satisfying the compatibility relation ordp(ρ_{1}) +ordp(ρ_{2})≥2g−2.

For every curveC of compact type, the varietyQ^{r}_{d}(C) of (gp)^{r}_{d}-relations has an
obvious determinantal scheme structure. One can construct a moduli stack of(gp)^{r}_{d}-
relations which has a natural determinantal structure over the moduli stack of curves of
compact type. In particular one has a lower bound on the dimension of each irreducible
component of this space and we shall use this feature in order to smooth(gp)^{r}_{d}-relations
constructed over curves from the divisor∆_{1} to nearby smooth curves fromM_{g}. The
proof of the following theorem is very similar to the proof of Theorem 4.3 in [F2] which
dealt with the caser= 1. We omit the details.

**Theorem 1.2.** *We fix integers*g, r, d*such that*ρ(g, r, d) ≥0*and a curve*[C := C_{1}∪y C_{2}]∈
Mg*of compact type. We denote by*π:C →B*the versal deformation space of*C =π^{−1}(0), with
0∈ B*. Then there exists a quasi-projective variety*ν :Q^{r}_{d}→ B*, compatible with base change,*
*such that the fibre over each point*b∈B*parameterizes*(gp)^{r}_{d}*-relations over*C_{b}*. Moreover, each*
*irreducible component of*Q^{r}_{d}*has dimension at least*dim(B)−1 = 3g−4*.*

The dimensional estimate onQ^{r}_{d}comes from its construction as a determinantal
variety overB. Just like in the case ofUe_{g,d}^{r} , we denote byǫ:Q^{r}_{d}→Ge^{r}_{d}the forgetful map
such thatσ◦ǫ=ν. We use the existence ofQ^{r}_{d}to prove the following inductive result:

**Theorem 1.3.** *Fix integers*g, r, d*such that*ρ(g, r, d) ≥ 2*and let us assume that*GP^{r}_{g,d}*has a*
*divisorial component*D*in*Mg *such that if*[C]∈ D*is a general point, then the variety*Q^{r}_{d}(C)

*has at least one*0-dimensional component corresponding to two complementary base point free
*linear series*(l, m) ∈ G^{r}_{d}(C)×G^{g−d+r−1}_{2g−2−d} (C), such that[C, l] ∈ Ge^{r}_{d}*is a smooth point. Then*
GP^{r}_{g+1,d+1} *has a divisorial component*D^{′}*in*Mg+1*such that a general point*[C^{′}]∈ D^{′} *enjoys*
*the same properties, namely that*Q^{r}_{d+1}(C^{′})*possesses a*0*-dimensional component corresponding*
*to a pair of base point free complementary linear series*(l^{′}, m^{′}) ∈ G^{r}_{d+1}(C^{′})×G^{g−d+r−1}_{2g−1−d} (C^{′})
*such that*[C^{′}, l^{′}]∈Ge^{r}_{d+1}*is a smooth point.*

*Proof.* We choose a general curve[C] ∈ D ⊂ GP^{r}_{g,d}, a general pointp ∈ C and we set
[C_{0} := C∪p E]∈ Mg+1, whereE is an elliptic curve. By assumption, there exist base
point free linear seriesl_{0} = (L, V) ∈ G^{r}_{d}(C) andm_{0} = (K_{C} ⊗L^{∨}, W) ∈ G^{s−1}_{2g−2−d}(C),
together with an element ρ ∈ **PKer** µ_{0}(V, W)

such that dim(l0,m0,ρ)Q^{r}_{d}(C) = 0. In
particular, then Kerµ_{0}(V, W)is1-dimensional. Letπ:C →Bbe the versal deformation
space of C_{0} = π^{−1}(0) and ∆ ⊂ B the boundary divisor corresponding to singular
curves. We consider the schemeν : Q^{r}_{d+1} → B parameterizing (gp)^{r}_{d+1}-relations (cf.

Theorem 1.2). Since[C, l_{0}]∈G^{r}_{d}is a smooth point andl_{0} is base point free, Lemma 2.5
from [AC] implies thatµ1(V, W) : Ker µ0(V, W) → H^{0}(K_{C}^{⊗2})is injective, in particular
µ_{1}(V, W)(ρ) 6= 0. (Hereσ_{0} :G^{r}_{d}→ Mg denotes the stack ofg^{r}_{d}’s over the moduli space
of curves of genusg). Thus we can assume that ordp(ρ) = 1for a generic choice ofp.

We construct a(gp)^{r}_{d+1}-relationz = (l, m, ρC, ρE) ∈ Q^{r}_{d+1}(C0)as follows: theC-
aspect of the limit g^{r}_{d+1} denoted bylis obtained by addingpas a base point to(L, V),
that is l_{C} = L_{C} := L ⊗ OC(p), V_{C} := V ⊂ H^{0}(L_{C})

. The aspect l_{E} is constructed
by adding(d−r)·pas a base locus to|L^{0}_{E}|, whereL^{0}_{E} ∈ Pic^{r+1}(E) is such thatL^{0}_{E} 6=

O_{E}((r + 1)·p) and(L^{0}_{E})^{⊗2} = O_{E}((2r + 2)·p), and where |V_{E}| = (d−r)·p+|L^{0}_{E}|.

Sincep ∈ C is general, we may assume thatpis not a ramification point of l_{0}, which
implies thata^{l}^{C}(p) = (1,2, . . . , r+ 1). Clearly,a^{l}^{E}(p) = (d−r, d−r+ 1,· · ·, d), hence
l = {lC, l_{E}}is a refined limitg^{r}_{d+1} onC_{0}. TheC-aspect of the limitg^{s−1}_{2g−2−d} we denote
by m, is given by m_{C} := K_{C} ⊗L^{∨} ⊗ OC(p), W_{C} := W ⊂ H^{0}(K_{C} ⊗L^{∨} ⊗ OC(p))

.
The aspect m_{E} is constructed by adding (g−r −1)·p to the complete linear series

|OE((r+ 1 +s)·p)⊗(L^{0}_{E})^{∨}|. Since we may also assume thatpis not a ramification point
ofm_{0}, we find thata^{m}^{C}(p) = (1,2, . . . , s)anda^{m}^{E}(p) = (g−r−1, g−r, . . . ,2g−2−d),
that is,m={mC, m_{E}}is a refined limitg^{s−1}_{2g−1−d}onC_{0}. Next we construct the elements
ρ_{C} andρ_{E}. We choose

ρ_{C} =ρ∈**PKer{µ**0(V, W) :V ⊗W →H^{0}(K_{C} ⊗ OC(2p))},

that is,ρ_{C} equalsρexcept that we addpas a simple base point to both linear seriesl_{C}
andm_{C} whose sections get multiplied. Clearly ordp(ρ_{C}) = ordp(ρ) + 2 = 3. Then we
construct an elementρ_{E} ∈ **PKer{V**E ⊗W_{E} → H^{0}(OE(2g·p))} with the property that
ordp(ρ_{E}) = 2g−3 = d−1 + (2g−2−d) = d+ (2g−3−d)

. Such an element lies necessarily in the kernel of the map

H^{0} L^{0}_{E}⊗ O_{E}(−(r−1)·p)

⊗H^{0} O_{E}((r+ 3)·p)⊗(L^{0}_{E})^{∨}

→H^{0}(O_{E}(4·p)),
which by the base point free pencil trick is isomorphic to the 1-dimensional space
H^{0} E,OE((2r + 2)·p)⊗(L^{0}_{E})^{⊗(−2)}

, that is, ρ_{E} is uniquely determined by the prop-
erty that ordp(ρE)≥2g−3.

Since ordp(ρC) +ordp(ρE) = 2g, we find thatz = (l, m, ρC, ρE) ∈ Q^{r}_{d+1}. The-
orem 1.2 guarantees that any component ofQ^{r}_{d+1} passing throughz has dimension at
least3g−1. To prove the existence of a component ofQ^{r}_{d+1} mapping rationally onto
a divisorD^{′} ⊂ Mg+1, it suffices to show thatzis an isolated point in ν^{−1}([C_{0}]). Sup-
pose thatz^{′} = (l^{′}, m^{′}, ρ^{′}_{C}, ρ^{′}_{E}) ∈ Q^{r}_{d+1} is another point lying in the same component of
ν^{−1}([C_{0}])asz. Since the schemeQ^{r}_{d+1} is constructed as a disjoint union over the pos-
sibilities of the vanishing sequences of the limit linear seriesg^{r}_{d+1}andg^{s−1}_{2g−1−d}, we may
assume thata^{l}^{′}^{C}(p) = a^{l}^{C}(p) = (1,2, . . . , r+ 1), a^{m}^{′}^{C}(p) = a^{m}^{C}(p) = (1,2, . . . , s). Sim-
ilarly for theE-aspects, we assume that a^{l}^{′}^{E}(p) = a^{l}^{E}(p) anda^{m}^{′}^{E}(p) = a^{m}^{E}(p). Then
necessarily, ordp(ρ^{′}_{C}) = 3(= 1 + 2 = 2 + 1), otherwise we would contradict the as-
sumptionµ_{1}(V, W)(ρ) = 0. Moreover, l_{C} = l_{0} andm_{C} = m_{0} because of the inductive
assumption on[C]. Using the compatibility relation betweenρ^{′}_{C}andρ^{′}_{E}we then get that
ordp(ρ^{′}_{E})≥2g−3. The only way this can be satisfied is when the underlying line bun-
dleL^{′}_{E} of the linear seriesl^{′}_{E}(−(d−r)·p)satisfies the relation(L^{′}_{E})^{⊗2} =OE((2r+ 2)·p),
which gives a finite number of choices forl^{′}_{E} and then form^{′}_{E}. Oncel^{′}_{E} is fixed, then as
pointed out before,ρ^{′}_{E} is uniquely determined by the condition ordp(ρ^{′}_{E})≥2g−3(and
in fact one must have equality). This shows thatz∈ν^{−1}([C_{0}])is an isolated point, thus
zmust smooth to(gp)^{r}_{d+1}relations on smooth curves filling-up a divisorD^{′}inMg+1.

We now prove that [C_{0}, l] ∈ Ge^{r}_{d+1} is a smooth point (Recall that σ : Ge^{r}_{d+1} → B
denotes the stack of limitg^{r}_{d+1}’s on the fibres ofπ). This follows once we show that[C_{0}, l]

is a smooth point ofσ^{∗}(∆)and then observe thatGe^{r}_{d+1} commutes with base change. By
explicit description, a neighbourhood of[C0, l]∈σ^{∗}(∆)is locally isomorphic to an ´etale
neighbourhood of(G^{r}_{d}×MgMg,1)× M1,1around the point [C, l0],[C, y],[E, y]

and we
can use our inductive assumption thatG^{r}_{d}is smooth at the point[C, l_{0}].

Finally, we prove that a generic point[C^{′}]∈ D^{′}corresponds to a pair of base point
free linear series(l^{′}, m^{′}) ∈ G^{r}_{d+1}(C^{′})×G^{s−1}_{2g−1−d}(C^{′}). Suppose this is not the case and
assume that, say,l^{′} ∈G^{r}_{d+1}(C^{′})has a base point. As[C^{′}, l^{′}]∈Ge^{r}_{d+1}specializes to[C_{0}, l_{0}]
the base point ofl^{′} specializes to a pointy ∈(C_{0})_{reg}(If the base point specialized to the
p ∈ C∩E, then necessarilylwould be a non-refined limitg^{r}_{d+1}). If y ∈ C − {p}then
it follows thatl_{0} = l_{C}(−p) ∈ G^{r}_{d}(C) has a base point aty, which is a contradiction. If
y∈E− {p}, thenL^{0}_{E} must have a base point atywhich is manifestly false.

2. THE CLASS OF THEGIESEKER-PETRI DIVISORS.

In this section we determine the class of the Gieseker-Petri divisor GP^{r}_{g,d}. We
start by setting some notation. We fix integers r, s ≥ 1 and setg := rs+s+ 1 and
d:=rs+r+1, henceρ(g, r, d) = 1. We denote byM^{0}_{g}the open substack ofMgconsisting
of curves[C]∈ M_{g} such thatW_{d}^{r+1}(C) =∅. Sinceρ(g, r+ 1, d) =−r−s−1, it follows
that codim(Mg − M^{0}_{g},Mg) ≥ 3. In this section we denote by G^{r}_{d} ⊂ Pic^{d} the stack
parameterizing pairs[C, l]with[C]∈ M^{0}_{g} andl ∈G^{r}_{d}(C)andMfg := M^{0}_{g}∪(∪^{[g/2]}_{i=0} ∆^{0}_{i}).

We have a natural projectionσ:G^{r}_{d}→ M^{0}_{g}. Furthermore, we denote byπ:M^{0}_{g,1} → M^{0}_{g}
the universal curve and byf :M^{0}_{g,1}×_{M}^{0}_{g} G^{r}_{d}→G^{r}_{d}the second projection. Note that the

forgetful mapǫ:U_{g,d}^{r} →G^{r}_{d}is an isomorphism overM^{0}_{g}, and we make the identification
betweenU_{g,d}^{r} andG^{r}_{d}(This identification obviously no longer holds overMf_{g}− M^{0}_{g}).

From general Brill-Noether theory it follows that there exists a unique component of
G^{r}_{d}which maps ontoM^{0}_{g}. Moreover, any irreducible componentZ ofG^{r}_{d}of dimension

>3g−3 +ρ(g, r, d)has the property that codim σ(Z),M^{0}_{g}

≥2(see [F1], Corollary 2.5 for a similar statement whenρ(g, r, d) = 0, the proof remains essentially the same in the caseρ(g, r, d) = 1).

If L is a Poincar´e bundle over M^{0}_{g,1} ×_{M}^{0}_{g} G^{r}_{d} (one may have to make an ´etale
base change Σ → G^{r}_{d} to ensure the existence ofL, see [Est]), we set F := f∗(L) and
N :=R^{1}f_{∗}(L). By Grauert’s theorem, bothF andN are vector bundles overG^{r}_{d}=U_{g,d}^{r}
with rank(F) =r+1and rank(N) =srespectively, and there exists a bundle morphism
µ:F ⊗ N →σ^{∗}(E), which over each point[C, L]∈G^{r}_{d}restricts to the Petri mapµ_{0}(L).
If U := Z_{rs+s−1}(µ) is the first degeneration locus of µ, then clearly GP^{r}_{g,d} = σ_{∗}(U).

Each irreducible component ofU has codimension at most2insideG^{r}_{d}. We shall prove
that every such component mapping onto a divisor in Mg is in fact of codimension
2 (see Proposition 2.3), which will enable us to use Porteous’ formula to compute its
class. While the construction ofFandN clearly depends on the choice of the Poincar´e
bundleL(and ofΣ), it is easy to check that the degeneracy classZ_{rs+s−1}(µ) ∈A^{2}(G^{r}_{d})
is independent of such choices.

Like in [F1], our technique for determining the class of the divisor GP^{r}_{g,d} is to
intersectUwith pull-backs of test curves sitting in the boundary ofMg: We fix a general
pointed curve[C, q]∈ M_{g−1,1}and a general elliptic curve[E, y]∈ M_{1,1}. Then we define
the families

C^{0}:={C/y∼q :y∈C} ⊂∆0 ⊂ Mg andC^{1}:={C∪yE:y∈C} ⊂∆1 ⊂ Mg.
These curves intersect the generators of Pic(Mg)as follows:

C^{0}·λ= 0, C^{0}·δ_{0} =−2g+ 2, C^{0}·δ_{1} = 1andC^{0}·δ_{j} = 0for2≤j≤[g/2], and
C^{1}·λ= 0, C^{1}·δ_{0}= 0, C^{1}·δ_{1} =−2g+ 4andC^{1}·δ_{j} = 0for2≤j ≤[g/2].

Next we fix a genus[g/2] ≤ j ≤ g−2and general curves[C] ∈ Mj,[D, y]∈ Mg−j,1.
We define the1-parameter familyC^{j} :={Cy^{j} =C∪yD}y∈C ⊂∆_{j} ⊂ Mg. We have the
formulas

C^{j} ·λ= 0, C^{j}·δ_{j} =−2j+ 2andC^{j}·δ_{i}= 0 fori6=j.

To understand the intersectionsC^{j} · GP^{r}_{g,d}for0≤j ≤[g/2], we shall extend the vector
bundlesF andN over the partial compactificationUe_{g,d}^{r} constructed in Section 1.

The next propositions describe the pull-back surfacesσ^{∗}(C^{j})insideGe^{r}_{d}:

**Proposition 2.1.** *We set*g:=rs+s+1*and fix general curves*[C]∈ Mrs+s*and*[E, y]∈ M1,1

*and consider the associated test curve*C^{1} ⊂∆_{1} ⊂ M_{g}*. Then we have the following equality of*
2*-cycles in*Ge^{r}_{d}*:*

σ^{∗}(C^{1}) =X+X_{1}×X_{2}+ Γ_{0}×Z_{0}+n_{1}·Z_{1}+n_{2}·Z_{2}+n_{3}·Z_{3},
*where*

X:={(y, L)∈C×W_{d}^{r}(C) :h^{0}(C, L⊗ OC(−2y)) =r}

X1 :={(y, L)∈C×W_{d}^{r}(C) :h^{0}(L⊗ OC(−2·y)) =r, h^{0}(L⊗ OC(−(r+ 2)·y)) = 1}

X_{2}:={(y, l)∈G^{r}_{r+2}(E) :a^{l}_{1}(y)≥2, a^{l}_{r}(y)≥r+ 2} ∼=* P*H

^{0}(O

_{E}((r+ 2)·y)) H

^{0}(OE(r·y))

Γ_{0} :={(y, A⊗ OC(y)) :y∈C, A∈W_{d−1}^{r} (C)}, Z_{0}=G^{r}_{r+1}(E) = Pic^{r+1}(E)
Z1 :={l∈G^{r}_{r+3}(E) :a^{l}_{1}(y)≥3, a^{l}_{r}(y)≥r+ 3} ∼=**P**

H^{0}(OE((r+ 3)·y))
H^{0}(OE(r·y))

Z_{2} :={l∈G^{r}_{r+2}(E) :a^{l}_{2}(y)≥3, a^{l}_{r}(y)≥r+ 2} ∼=* P*H

^{0}(OE((r+ 2)·y)) H

^{0}(OE((r−1)·y))

Z3:={l∈G^{r}_{r+2}(E) :a^{l}_{1}(y)≥2}= [

z∈E

**P**

H^{0}(OE((r+ 1)·y+z))
H^{0}(O_{E}((r−1)·y+z))

,
*where the constants*n_{1}, n_{2}, n_{3}*are explicitly known positive integers.*

*Proof.* Every point inσ^{∗}(C^{1})corresponds to a limitg^{r}_{d}, sayl ={lC, l_{E}}, on some curve
[C_{y}^{1} := C ∪y E] ∈ C^{1}. By investigating the possible ways of distributing the Brill-
Noether numbersρ(l_{C}, y)andρ(l_{E}, y)in a way such that the inequality1 =ρ(g, r, d) ≥
ρ(lC, y) + ρ(lE, y) is satisfied, we arrive to the six components in the statement (We
always use the elementary inequalityρ(l_{E}, y)≥0, henceρ(l_{C}, y)≤1). We mention that
X corresponds to the case whenρ(lC, y) = 1, ρ(lE, y) = 0, the surfaces X1 ×X2 and
Γ_{0}×Z_{0} correspond to the caseρ(l_{C}, y) = 0, ρ(l_{E}, y) = 0, whileZ_{1}, Z_{2}, Z_{3}appear in the
cases whenρ(l_{C}, y) = −1, ρ(lE, y) = 1. The constants n_{i} for1 ≤ i ≤ 3have a clear
enumerative meaning: First, n_{1} is the number of pointsy ∈ C for which there exists
L ∈ W_{d}^{r}(C) such thata^{L}(y) = (0,2,3, . . . , r, r+ 3). Then n_{2} is the number of points
y∈Cfor which there existsL∈W_{d}^{r}(C)such thata^{L}(y) = (0,2,3, . . . , r−1, r+ 1, r+ 2).

Finally,n_{3}is the number of pointsy ∈Cwhich appear as ramification points for one of

the finitely many linear seriesA∈W_{d−1}^{r} (C).

Next we describeσ^{∗}(C^{0})and we start by fixing more notation. We choose a gen-
eral pointed curve[C, q]∈ Mrs+s,1and denote byY the following surface:

Y :={(y, L)∈C×W_{d}^{r}(C) :h^{0}(C, L⊗ OC(−y−q)) =r}.

Let π_{1} : Y → C denote the first projection. Inside Y we consider two curves corre-
sponding tog^{r}_{d}’s with a base point atq:

Γ_{1} :={(y, A⊗ O_{C}(y)) :y ∈C, A∈W_{d−1}^{r} (C)} and
Γ_{2} :={(y, A⊗ O_{C}(q)) :y∈C, A∈W_{d−1}^{r} (C)},

intersecting transversally inn_{0} := #(W_{d−1}^{r} (C))points. Note thatρ(g, r−1, d) = 0and
W_{d−1}^{r} (C)is a reduced0-dimensional cycle. We denote byY^{′}the blow-up ofY at thesen_{0}
points and at the points(q, B)∈Y whereB∈W_{d}^{r}(C)is a linear series with the property
thath^{0}(C, B⊗ OC(−(r+ 2)·q))≥1. We denote byE_{A}, E_{B} ⊂Y^{′}the exceptional divisors
corresponding to(q, A⊗ OC(q))and(q, B) respectively, byǫ : Y^{′} → Y the projection
and byΓe1,Γe2⊂Y^{′}the strict transforms ofΓ1andΓ2.

**Proposition 2.2.** *Fix a general curve*[C, q] ∈ Mrs+s,1 *and consider the associated test curve*
C^{0}⊂∆_{0} ⊂ M_{rs+s+1}*. Then we have the following equality of*2-cycles inGe^{d}_{r}*:*

σ^{∗}(C^{0}) =Y^{′}/eΓ_{1} ∼=Γe_{2},

*that is,*σ^{∗}(C^{0})*can be naturally identified with the surface obtained from*Y^{′} *by identifying the*
*disjoint curves*Γe1 *and*Γe2 *over each pair*(y, A)∈C×W_{d−1}^{r} (C).

*Proof.* We fix a pointy ∈ C − {q}, denote by [C_{y}^{0} := C/y ∼ q] ∈ Mg, ν : C → C_{0}^{y}
the normalization map, and we investigate the varietyW^{r}_{d}(C_{y}^{0}) ⊂ Pic^{d}(C_{y}^{0})of torsion-
free sheavesL onC_{y}^{0} with deg(L) = dand h^{0}(C_{y}^{0}, L) ≥ r+ 1. If L ∈ W_{d}^{r}(C_{y}^{0}), that
is, L is locally free, thenLis determined by ν^{∗}(L) ∈ W_{d}^{r}(C) which has the property
that h^{0}(C, ν^{∗}L⊗ OC(−y −q)) = r. However, the line bundles of typeA⊗ OC(y) or
A⊗ OC(q)withA∈W_{d−1}^{r} (C), do not appear in this association even though they have
this property. They correspond to the situation whenL ∈ W^{r}_{d}(C_{0}^{y}) is not locally free,
in which case necessarilyL = ν_{∗}(A) for someA ∈ W_{d−1}^{r} (C). ThusY ∩π_{1}^{−1}(y) is the
partial normalization ofW^{r}_{d}(C_{y}^{0})at then_{0} points of the formν_{∗}(A)withA ∈W_{d−1}^{r} (C).
A special analysis is required when y = q, that is, when C_{y}^{0} degenerates to C ∪_{q}E_{∞},
where E_{∞} is a rational nodal cubic. If {lC, l_{E}_{∞}} ∈ σ^{−1}([C ∪qE_{∞}]), then an analysis
along the lines of Theorem 2.1 shows thatρ(l_{C}, q) ≥ 0 andρ(l_{E}_{∞}, q) ≤ 1. Then either
l_{C} has a base point atqand then the underlying line bundle ofl_{C} is of typeA⊗ OC(q)
whilel_{E}_{∞}(−(d−r−1)·q)∈W^{r}_{r+1}(E_{∞}), or else,a^{l}^{C}(q) = (0,2,3, . . . , r, r+ 2)and then
l_{E}_{∞}(−(d−r−2)·q)∈**P** H^{0}(OE∞((r+ 2)·q))/H^{0}(OE∞(r·q))∼=E_{B}, whereB ∈W_{d}^{r}(C)

is the underlying line bundle ofl_{C}.

We now show that every irreducible component ofU has the expected dimension:

**Proposition 2.3.** *Every irreducible component* X *of* U *having the property that* σ(X) *is a*
*divisor in*M_{g}*has*codim(X,G^{r}_{d}) = 2.

*Proof.* Suppose thatX is an irreducible component ofUsatisfying (1) codim(X,G^{r}_{d})≤1
and (2) codim(σ(X),Mg) = 1. We writeD:=σ(X)⊂ Mgfor the closure of this divisor
inMg, and we express its class asD≡aλ−b_{0}δ_{0}−b_{1}δ_{1}− · · · −b_{[g/2]}δ_{[g/2]}∈Pic(Mg). To
reach a contradiction, it suffices to show thata= 0.

Keeping the notation from Propositions 2.1 and 2.2, we are going to show that
C^{0}∩D=C^{1}∩D=∅which implies thatb_{0}=b_{1}= 0. Then we shall show that ifR⊂ M_{g}
denotes the pencil obtained by attaching to a general pointed curve[C, q]∈ Mrs+s,1at
the fixed point q, a pencil of plane cubics (i.e. an elliptic pencil of degree 12), then
R∩D=∅. This implies the relationa−12b_{0}+b_{1}= 0which of course yields thata= 0.

We assume by contradiction that C^{1} ∩D 6= ∅. Then there exists a pointy ∈ C
and a limitg^{r}_{d}onC_{y}^{1} :=C∪y E, sayl ={lC, l_{E}}, such that ifL_{C} ∈ W_{d}^{r}(C)denotes the
underlying line bundle ofl_{C}, then the multiplication map

µ_{0}(L_{C}, y) :H^{0}(L_{C})⊗H^{0}(K_{C}⊗L^{∨}_{C}⊗ OC(2y))→H^{0}(K_{C}⊗ OC(2y))

is not injective. We claim that this can happen only whenρ(l_{C}, y) = 1andρ(l_{E}, y) = 0,
that is, when[C_{y}^{1}, l]∈X (we are still using the notation from Proposition 2.1). Indeed,