Loci of curves with subcanonical points in low genus

0.36 MB·English

by  Dawei Chen

LOCI OF CURVES WITH SUBCANONICAL POINTS IN LOW GENUS
6
DAWEICHENANDNICOLATARASCA
1
0
2 Abstract. Inside the moduli space of curves of genus three with one marked point, we con-

b sider the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-
e hyperelliptic curves with a marked hyperflex. These loci have codimension two. We compute
F theclassesoftheirclosuresinthemodulispaceofstablecurvesofgenusthreewithonemarked
point. Similarly, we compute the class of the closure of the locus of curves of genus four with
4
aneven theta characteristic vanishingwithorder threeat acertainpoint. Theseloci naturally
2
ariseinthestudyofminimaldimensionalstrataofAbeliandifferentials.

]
G
A
1. Introduction
.
h
Apointponasmoothcurveofgenusg ≥2iscalledsubcanonicalif(2g−2)pisacanonicaldivisor.
t
a For example, Weierstrass points of a hyperelliptic curve are subcanonical points. Subcanonical
m
points are the most special Weierstrass points from the perspective of Weierstrass gap sequences
[ ([ACGH85, Exercise 1.E]). The locus of curves admitting a subcanonical point inside the moduli

space M of curves of genus g has codimension g−2.
2 g
v SuchlocinaturallyarisefromthestudyofstrataofAbeliandifferentials. LetHg bethe moduli
5 space of Abelian differentials parameterizing pairs (C,ω), where C is a smooth curve of genus g
3 and ω is an Abelian differential on C, for g ≥ 2. Given µ = (m ,...,m ) a partition of 2g−2,
1 n
2 denote by H(µ) ⊆ H the stratum of Abelian differentials (C,ω), where the zeros of ω are of
g
2 type µ,namely,(ω) = n m p ,fordistinctpoints p . ThereisaGL+(R)-actiononH(µ)that
0 0 i=1 i i i 2
. varies the realandimagPinaryparts of ω ([Zor06]). The study of GL+(R)-orbits is a major subject
1 2
in Teichmu¨ller dynamics, with fascinating applications to the geometry of the moduli space of
0
5 stable curves, such as producing rigid curves, and bounding slopes of effective divisors ([Che11]).
1 Moreover, one can degenerate (C,ω) to a nodal curve along with a section of the dualizing sheaf,
: that is, compactify H(µ) over the Deligne-Mumford moduli space M of stable nodal curves,
v g
i and study its boundary behavior. This turns out to provide crucial information for invariants in
X
Teichmu¨ller dynamics, such as Siegel-Veech constants, Lyapunov exponents, and classification of
r Teichmu¨ller curves ([CM12, BM12]).
a
AmongallstrataofAbeliandifferentials,theminimaldimensionalstratumisH(2g−2),param-
eterizingpairs(C,ω),where(ω) =(2g−2)pforacertainpointpinC. Notethatif(ω) =(2g−2)p,
0 0
then O ((g −1)p) is a spin structure on C, that is, a square root of the canonical bundle. The
C
parity of a spin structure η on a curve C is defined as the parity of h0(C,η).
For g ≥4, the stratumH (2g−2) has three connected components: the componentof hyperel-
lipticcurvesH(2g−2)hyp,andthetwocomponentsH(2g−2)evenandH(2g−2)odddistinguished
by the parity of the corresponding spin structures. For g = 3, the hyperelliptic and even compo-
nents coincide, hence H(4) has two connected components: H(4)hyp and H(4)odd. For g = 2,
2010 Mathematics Subject Classification. 14H99(primary),14C99(secondary).
Keywordsandphrases. Higher Weierstrass points, spin curves, minimal strata of Abelian differentials,
effectivecyclesinmodulispacesofcurves.
During the preparation of this article the first author was partially supported by NSF grant DMS-1200329
andNSFCAREERgrantDMS-1350396.
1
2 DAWEICHENANDNICOLATARASCA
the stratum H(2) is irreducible. The reader can refer to [KZ03] for a complete classification of
connected components of H(µ), where the minimal dimensional stratum H(2g − 2) plays an
important role as a base case for an inductive argument.
One can project H(µ) to M via the forgetful map (C,ω) 7→ C. Alternatively, by marking
g
the zeros of ω, one can lift H (µ) (up to rescaling ω) to M . Thus, one obtains a number of
g,n
interesting subvarieties in M and M . For example, the projection of H(2) dominates M ,
g g,n 2
and the lift of H (2) is the divisor of Weierstrass points in M .
2,1
Motivated by the problem of determining the Kodaira dimension of moduli spaces of stable
curves M , effective divisor classes in M have been extensively calculated in the last several
g,n g,n
decades. In contrast, much less is known about higher codimensional cycles in M . Recently,
g,n
there has been growing interest in studying higher codimensional subvarieties of moduli spaces
of curves ([FP05, Hai13, FP15, Tar13, Tar15]), and in describing cones of higher codimensional
effective cycles ([CC15]). Our main result is the explicit computation of classes of closures of loci
of curves with subcanonical points in the moduli space of stable curves in low genus.
Let Hyp be the locus of smooth hyperelliptic curves with a marked Weierstrass point in
3,1
M . Let F be the locus of smooth non-hyperelliptic curves with a marked hyperflex point in
3,1 3,1
M . The loci Hyp and F have codimension two in M . They are the lifts of the minimal
3,1 3,1 3,1 3,1
dimensional stratum components H(4)hyp and H (4)odd, respectively. In §4 and §5 we prove the
following result in the Chow group A2(M ) of codimension-two classes on M .
3,1 3,1
Theorem 1.1. The classes of the closures of Hyp and F in A2(M ) are
3,1 3,1 3,1
19 1 5
Hyp ≡ ψ(18λ−2δ −9δ −6δ )−λ 45λ− δ −24δ − δ2− δ δ −3δ2 ,
3,1 0 1,1 2,1 (cid:18) 2 0 2,1(cid:19) 2 0 2 0 2,1 2,1
137 7
F ≡ ψ(77λ−3ψ−8δ −42δ −19δ )−λ 338λ− δ −146δ − δ2
3,1 0 1,1 2,1 (cid:18) 2 0 2,1(cid:19) 2 0
31
− δ δ −3δ2 −20δ2 +3κ .
2 0 2,1 1,1 2,1 2
As a check, consider the map p: M → M obtained by forgetting the marked point. The
3,1 3
push-forwardof products of divisor classes in M via p are described in §5.3. It follows that the
3,1
push-forward of the class of Hyp from Theorem 1.1 via p is
3,1
p Hyp ≡8·Hyp ,
∗ 3,1 3
(cid:0) (cid:1)
as expected, where Hyp is the closure of the locus of hyperelliptic curves in M . Indeed, from
3 3
[HM82]wehaveHyp ≡9λ−δ −3δ ,andthemultiplicity8accountsforthenumberofWeierstrass
3 0 1
pointsinahyperellipticcurveofgenus3. Theclassofthedivisorp (F )iscomputedin[Cuk89],
∗ 3,1
and will be used in the proof of Theorem 1.1.
Similarly, we can consider the following codimension-two loci in the moduli space of curves of
genus 4. Let H be the locus in M of smooth curves C that admit a canonical divisor of type
4 4
6p. It consists of the three irreducible components Hyp , H+ and H−, defined as the projection
4 4 4
imagesoftheminimaldimensionalstratumcomponentsH (6)hyp,H(6)even,andH(6)odd toM ,
4
respectively.
TheclassoftheclosureofHyp hasbeenfirstcomputedin[FP05,Proposition5]vialocalization
4
on the moduli space of stable relative maps (see (5)). An alternative proof via test surfaces and
admissible covers is given in [Tar13]. In §6 we obtain the following result in the Chow group
A2(M ).
4
LOCI OF CURVES WITH SUBCANONICAL POINTS IN LOW GENUS 3
Theorem 1.2. The class of the closure of the locus H+ in A2(M ) is
4 4
H+ ≡ 2448λ2−542λδ −1608λδ +276λδ +32δ2+178δ δ
4 0 1 2 0 0 1
+336δ2+276δ δ +576δ2−4δ −60γ +12δ −144δ .
1 1 2 2 00 1 01a 1|1
As an immediate consequence of Theorems 1.1 and 1.2, in §7 we prove the following geometric
result.
+
Corollary 1.1. The loci Hyp , F , and H are not complete intersections in their respective
3,1 3,1 4
spaces.
The class of Hyp is known to span an extremal ray of the cone of effective codimension-two
3,1
classes in M (similarly for the locus Hyp in M ) ([CC15]). It is natural to ask whether the
3,1 4 4
+
classes of F and H lie in the interior or in the boundary of the cones of effective codimension-
3,1 4
+
two classes of their respective spaces. In addition, the loci Hyp , F , and H are images of
3,1 3,1 4
the strata H(4)hyp, H(4)odd, and H(6)even in the respective moduli spaces of curves. Thus our
results can shed some light on the study of cones of effective higher codimensional cycles, as well
as the study of degenerate Abelian differentials appearing in the boundary of these strata.
Letusdescribeourmethods. InordertostudytheclosureoftheimagesofthestrataofAbelian
differentials inside moduli spaces of curves, one needs a good description of singular hyperelliptic
curves, and singular curves with an even or odd spin structure. The closure of the hyperelliptic
components can be studied via the theory of admissible covers ([HM82]). Moreover, let S+ (re-
g
spectively, S−) be the moduli space of pairs [C,η], where C is a smooth curve of genus g, and
g
η is an even (respectively, odd) theta characteristic on C, that is, a spin structure on C. There
is a natural map S± → M obtained by forgetting the theta characteristic. We use Cornalba’s
g g
±
description of the closure S →M ([Cor89], see also [Far11, Far10, FV14]), and realize the loci
g g
+ − +
F and H as push-forwardof loci in S × M and S , respectively.
3,1 4 3 M3 3,1 4
Toperformourcomputations,weusethebasisforA2(M )from[Fab90b]. Itisnotclearwhether
4
the group of tautological classes R2(M ) ⊆ A2(M ) coincides with A2(M ). Anyhow, it is
3,1 3,1 3,1
a consequence of the descriptions in Lemmata 4.1 and 5.2 that the classes of Hyp and F
3,1 3,1
are tautological. In §3, we describe a basis for R2(M ). The classes ψ,λ,δ ,δ ,δ form a
3,1 0 1,1 2,1
Q-basis of Pic(M ). Note that the product λδ is linearly dependent from the other products,
3,1 1,1
see Proposition 3.1.
Proposition 1.1. The following 16 classes
(1) ψ2,ψλ,ψδ ,ψδ ,ψδ ,λ2,λδ ,λδ ,δ2,δ δ ,δ δ ,δ2 ,δ δ ,δ2 ,δ ,κ
0 1,1 2,1 0 2,1 0 0 1,1 0 2,1 1,1 1,1 2,1 2,1 01a 2
form a Q-basis of R2(M ).
3,1
Thepaperisorganizedasfollows. Wecollectin§2someresultsabouttheenumerativegeometry
of a general curve. In §3, we prove Proposition 1.1 using the following strategy. After [GL99,
Yan10],itisknownthatthegroupR2(M )hasdimension16,equaltotherankoftheintersection
3,1
pairing R2(M )×R5(M ). Hence, one could show that the intersection pairing of classes in
3,1 3,1
(1) with classes of complementary dimension has rank 16. Instead, we reduce the number of
computations by using Getzler’s results on R2(M ) ([Get98]). Let ϑ: M →M be the map
2,2 2,2 3,1
obtained by attaching a fixed elliptic tail to the second marked point. We first show that the
pull-backs of the classes in (1) via ϑ span a 13-dimensional subspace of the 14-dimensional space
R2(M ). We compute the 3-dimensional space of possible relations among the classes in (1).
2,2
Finally, by restricting to three test surfaces, we verify that such relations cannot hold.
4 DAWEICHENANDNICOLATARASCA
In §4 we compute the class of Hyp by studying the pull-back of the class Hyp via the
3,1 4
map j : M → M obtained by attaching a fixed elliptic tail at the marked point of curves in
3 3,1 4
+
M . To compute the classes of F and H , in §5 and §6 we realize each one of these loci as a
3,1 3,1 4
component of the intersection of two divisors in their respective moduli spaces. We first describe
set-theoreticallytheothercomponentsintheintersections. Themultiplicityalongeachcomponent
is then computed by restricting to test surfaces, and by studying the push-forward via the map
+
p: M →M . TheclassesofF andH followfromthecomputationoftheclassesoftheother
3,1 3 3,1 4
components in each intersection. Throughout the paper, we provide various checks on Theorems
1.1 and 1.2 (see §3.4, (6), Remark 4.2, §5.4, (9)).
We have not succeeded in applying the above ideas to obtain a complete formula for the class
− −
of H . One can directly intersect the locus H with some test surfaces and thus obtain linear
4 4
relations on the coefficients of the class. While in this way we have produced some relations for
suchcoefficients,atthemomentwehavenotsucceededincomputingthewholeclass. Nevertheless,
in§8.1wecomputetheclassofH− inA2(M )(thatis,thecoefficientofλ2)usingadeterminantal
4 4
description.
Notation. We use throughout the following notation for divisor classes in Pic(M ). For i =
g,n
1,...,n, let ψ be the cotangent line bundle class at the i-th marked point. Let λ be the first
i
Chern class of the Hodge bundle, and δ be the class of the locus ∆ whose general element is a
0 0
nodal irreducible curve. For i=0,...,g and S ⊆{1,...,n}, let δ be the class of the locus ∆
i,S i,S
whose general element has a component of genus i containing the points with markings in S, and
meeting transversally in one point a component of genus g−i containing the remaining marked
points. We write δ := δ and ∆ := ∆ . When n = 1, we write ψ := ψ , δ := δ , and
i i,∅ i i,∅ 1 i,1 i,{1}
∆ :=∆ .
i,1 i,{1}
We also use the following codimension-two classes. Let κ := (p ) ((ψ )3), where
2 n+1 ∗ n+1
p : M → M is the natural map obtained by forgetting the last marked point. Let
n+1 g,n+1 g,n
δ be the class of the closure of the locus of irreducible curves with two nodes. Let γ be the
00 i,S
class of the closure of the locus Γ of curves with a component of genus i containing the points
i,S
with markings in S and meeting in two points a component of genus g −i−1 containing the
remaining marked points. We write γ := γ and Γ := Γ . Let δ be the class of the closure
i i,∅ i i,∅ 01a
ofthe locus of curveswith a rationalnodaltail. Finally, δ is the class ofthe closureof the locus
1|1
of curves with two unmarked elliptic tails.
We work over an algebraically closed field of characteristic 0. All cycle classes are stack funda-
mental classes, and all cohomology and Chow groups are taken with rational coefficients.
Acknowledgements. We would like to thank Izzet Coskun, Joe Harris, Scott Mullane, and
Anand Patel for helpful conversations on related topics. We are grateful to the referee for many
valuable suggestions for improving the exposition of the paper.
2. Enumerative geometry of general curves
In this section, we collect some results about the enumerative geometry of general curves for
later use.
2.1. The difference map. Let C be a general curve of genus two. Define a generalized Abel-
Jacobi map f : C×C →Pic1(C) as follows:
d
(x,y)7→O ((d+1)x−dy),
C
where d∈Z+. Let ∆ ⊂C×C be the diagonal and I ⊂C×C be the locus of pairs of points
C×C
that are conjugate under the hyperelliptic involution. The intersection of I and ∆ consists of
C×C
the six Weierstrass points of C ∼=∆C×C.
LOCI OF CURVES WITH SUBCANONICAL POINTS IN LOW GENUS 5
Proposition 2.1. Under the above setting, we have:
(1) f is finite of degree 2d2(d+1)2;
d
(2) f is simply ramified along ∆ and I, away from I∩∆ ;
d C×C C×C
(3) The ramification order of f is 2 at each point in the intersection I∩∆ .
d C×C
Proof. Take a general point p ∈ C. Consider the isomorphism u: Pic1(C) → J(C) given by
L7→L⊗O (−p). Leth =u◦f . Notethath (x,y)=O ((d+1)(x−p)−d(y−p)). Letθ be the
C d d d C
fundamental class of the theta divisor in J(C). For k ∈ Z, the locus of O (k(x−p)) for varying
C
x∈C has class k2θ in J(C). Therefore, we conclude that
degf =degh =degh h∗([O ])=deg (d+1)2θ·d2θ =2d2(d+1)2,
d d d∗ d C
(cid:0) (cid:1)
thus proving the degree part of the proposition.
Next, let φ: C →P1 be the hyperelliptic double covering. By [ACGH85, p. 262],the associated
(projectivized) tangent space map of f at (x,y) can be regarded as P1 → hφ(x),φ(y)i, where
d
hφ(x),φ(y)i is the linear span of φ(x) and φ(y) in P1. Therefore, f is ramified at (x,y) if and
d
only if φ(x) = φ(y), that is, (x,y) ∈ ∆ ∪I. In particular, f is finite away from ∆ and
C×C d C×C
I. Moreover, f (w,w) = O (w) for any Weierstrass point w ∈ C, hence f (∆ ) and f (I)
d C d C×C d
cannot be a single point. We thus conclude that f is finite. Finally, via a local computation in a
d
neighborhoodof a Weierstrasspoint of C, one verifies the ramificationordersalong ∆ , I, and
C×C
∆ ∩I. (cid:3)
C×C
Similarly,weconsiderthefollowingmap. LetpbeafixedpointonageneralcurveCofgenustwo.
Fix two positive integers d ,d , and let d=d −d −1. Define the map f : C×C →Picd(C)
1 2 1 2 d1,d2
by
(x,y)7→O (d x−d y−p).
C 1 2
The same proof as in Proposition 2.1 implies the following result.
Proposition 2.2. The degree of f is 2d2d2.
d1,d2 1 2
2.2. The Scorza curve. Let C be a curve of genus g, and η+ an even theta characteristic on C.
Suppose that η+ is not a vanishing theta-null, that is, h0(C,η+) = 0. The Scorza curve T in
η+
C×C is defined as
T :={(x,y)∈C×C|h0(η+⊗O(x−y))6=0}
η+
(see [Sco00]). From the assumption, T does not meet the diagonal ∆ ⊂ C ×C. By the
η+ C×C
Riemann-Roch theorem, T is a symmetric correspondence.
η+
We need the following computation: when T is reduced, its class in H2(C×C) is
η+
T =(g−1)F +(g−1)F +∆ ,
η+ 1 2 C×C
(cid:2) (cid:3)
where F and F are the classes of a fiber of the projections C ×C → C of the first and second
1 2
components,respectively(see[DK93,§7.1]). Wealsousethatforageneralspincurve[C,η+]∈S+
g
with g ≥2, the Scorza curve T is smooth. In particular, the locus of pairs (x,y) ∈C×C such
η+
that
h0(C,η+⊗O(x−2y))≥1 and h0(C,η+⊗O(y−2x))≥1
is empty (see for instance [FV14, Theorem 4.1]).
3. Tautological classes on M
3,1
In this section, we show that the classes in (1) form a basis of R2(M ), and thus prove
3,1
Proposition 1.1. Moreover, in §3.3 we express certain boundary strata classes in terms of such a
basis. We use these results in §4 and §5.
6 DAWEICHENANDNICOLATARASCA
3.1. Proof of Proposition 1.1. From [GL99] we know that the dimension of H4(M ) is 16
3,1
(see also[Ber08]). Since the intersectionpairingofclassesinthe tautologicalgroupRH2(M )⊆
3,1
H4(M )withtautologicalclassesofcomplementarydimensionhasrank16(see[Yan10]),wecon-
3,1
clude that RH2(M ) = H4(M ). Every relation among codimension-two tautological classes
3,1 3,1
in RH2(M ) lifts via the cycle map to R2(M ), hence R2(M )≃RH2(M )=H4(M ).
3,1 3,1 3,1 3,1 3,1
In order to prove that the classes in (1) form a basis, it is enough to show that the intersection
pairing of the classes in (1) with classes in complementary dimension has rank 16. This is the
strategy used by Getzler for H∗(M ) in [Get98].
2,2
Here we use a different approach, relying on Getzler’s results. Remember that R2(M ) has
2,2
dimension 14, and one has R2(M )≃RH2(M )=H4(M ). The idea is the following. First
2,2 2,2 2,2
we consider the natural map ϑ: M → M obtained by attaching a fixed elliptic tail to the
2,2 3,1
secondmarkedpoint. Weshowthatthepull-backsoftheclassesin(1)viaϑspana13-dimensional
subspaceofR2(M ). Whencewecomputethepossiblethree-dimensionalspaceofrelationswhich
2,2
may still hold among the above classes, depending on three parameters α,β,γ. Finally, using test
surfaces we conclude that α=β =γ =0.
A basis for the Picard group of M is given by the following divisor classes
2,2
ψ , ψ , δ , δ , δ , δ .
1 2 0,{1,2} 0 1,{1} 1,{1,2}
Getzler shows that the following relations hold among products of divisor classes
δ 12δ +12δ +δ =δ 12δ +12δ +δ =0,
1,{1,2} 1,{1} 1,{1,2} 0 1,{1} 1,{1} 1,{1,2} 0
(cid:0) (cid:1) (cid:0) (cid:1)
δ ψ +ψ +δ =ψ δ =ψ δ =δ δ =0,
1,{1} 1 2 1,{1} 1 0,{1,2} 2 0,{1,2} 1,{1} 0,{1,2}
(cid:0) (cid:1)
(ψ −ψ ) 10ψ +10ψ −2δ −12δ −δ =0,
1 2 1 2 1,{1} 1,{1,2} 0
(cid:0) (cid:1)
hence a basis for R2(M ) is given by the following products
2,2
ψ ψ , ψ2, ψ δ , ψ δ , ψ δ , ψ δ , ψ δ , ψ δ ,
1 2 2 1 1,{1} 2 1,{1} 1 1,{1,2} 2 1,{1,2} 1 0 2 0
δ2 , δ δ , δ δ , δ δ , δ δ , δ2.
0,{1,2} 1,{1,2} 0,{1,2} 0 0,{1,2} 0 1,{1} 0 1,{1,2} 0
The following formulae are well-known
1 1
ϑ∗(ψ)=ψ , ϑ∗(λ)=λ= δ + (δ +δ ), ϑ∗(δ )=δ ,
1 10 0 5 1,{1} 1,{1,2} 0 0
ϑ∗(δ )=δ +δ , ϑ∗(δ )=−ψ +δ .
1,1 1,{1} 0,{1,2} 2,1 2 1,{1,2}
Moreover,from the computations in [Get98] it follows that
ϑ∗(δ ) = δ
01a 01a
54 114 12
(2) = 24ψ2−6ψ δ − ψ δ +6ψ δ − ψ δ − ψ δ
2 1 1,{1} 5 2 1,{1} 1 1,{1,2} 5 2 1,{1,2} 5 2 0
84 12 47 36 3
+24δ2 + δ δ + δ δ + δ δ + δ δ + δ2.
0,{1,2} 5 1,{1,2} 0,{1,2} 5 0 0,{1,2} 50 0 1,{1} 25 0 1,{1,2} 25 0
Getzler expresses the basis of R2(M ) given by products of divisor classes in terms of decorated
2,2
boundary strata classes. By the inverse change of basis, we obtain the formula (2).
It is easy to show that ϑ∗(κ )=κ . Note that by Mumford’s relation, one has
2 2
κ =λ(λ+δ )∈A2(M ).
2 1 2
LOCI OF CURVES WITH SUBCANONICAL POINTS IN LOW GENUS 7
Using the well-known formula κ = p∗(κ )+ψi, where p : M → M is the natural map,
i n i n n g,n g,n−1
we obtain that
(3) ϑ∗(κ )=κ = λ(λ+δ +δ )+ψ2+ψ2+δ2
2 2 1,{1} 1,{1,2} 1 2 0,{1,2}
3 1
= ψ2+ψ2+δ2 + δ (δ +δ )+ δ2 ∈A2(M ).
1 2 0,{1,2} 25 0 1,{1} 1,{1,2} 100 0 2,2
Using the above formulae, it is easy to compute that the pull-back via ϑ of the classes in the
statement span a subspace of R2(M ) of dimension 13. The only possible relations are given by
2,2
1 1 1
α· −2ψ2+6ψλ− ψδ −2ψδ −6λ2+ λδ +6λδ − δ δ −δ2 +κ
(cid:18) 2 0 1,1 2 0 2,1 2 0 2,1 1,1 2(cid:19)
1 1 1
+β· −ψ2+6ψλ− ψδ −5λ2+ λδ −4λδ + δ δ +δ2
(cid:18) 2 0 2 0 2,1 2 0 2,1 2,1(cid:19)
+γ· 120λ2−22λδ +δ2 =0
0 0
for α,β,γ ∈ Q. Restricting this relation to the three test surf(cid:0)aces in §3.2 yields α(cid:1)= β = γ = 0.
This completes the proof of the statement. (cid:3)
3.2. Three test surfaces in M . In the next subsections, we consider the intersections of the
3,1
classes in Proposition 1.1 with three test surfaces. These computations are used in the proof of
Proposition 1.1. Note that κ has zero intersection with the two-dimensional families of curves
2
whose base is a product of two curves.
3.2.1. Let (E ,p,q ) be a generaltwo-pointedelliptic curve, and let (E ,q ) be a pointed elliptic
1 1 2 2
curve. Consider the surface in M obtained by identifying the points q ,q , andby identifying a
3,1 1 2
moving point in E with a moving point in E .
1 2
Figure 1. How the general fiber of the family in §3.2.1 moves.
The base of the surface is E ×E =:S . Let π : E ×E →E be the natural projection, for
1 2 1 i 1 2 i
i=1,2. The divisor classes restrict as follows
ψ| =π∗[p], λ| =0, δ | =−2π∗[q ]−2π∗[q ], δ | =π∗[q ], δ | =π∗[q ].
S1 1 S1 0 S1 1 1 2 2 1,1 S1 1 1 2,1 S1 2 2
TheclassesinProposition1.1withnonzerointersectionwiththistestsurfacearethusthefollowing
ψδ | =−2, ψδ | =1, δ2| =8, δ δ | =−2, δ δ | =−2, δ δ | =1.
0 S1 2,1 S1 0 S1 0 1,1 S1 0 2,1 S1 1,1 2,1 S1
3.2.2. Let(E ,q ,r)and(E ,q ,p)betwogeneraltwo-pointedellipticcurves. Identifythepoints
1 1 2 2
q ,q ; identify the point r with a moving point in E ; finally move the curve E in a pencil of
1 2 1 2
degree 12.
ThebaseofthissurfaceisE ×P1 =:S . LetxbetheclassofapointinP1. Thedivisorclasses
1 2
restrict as follows
ψ| =π∗(x)=λ| , δ | =−2π∗[r]+12π∗(x), δ | =−π∗[q ]−π∗(x), δ | =π∗[r],
S2 2 S2 0 S2 1 2 1,1 S2 1 1 2 2,1 S2 1
8 DAWEICHENANDNICOLATARASCA
Figure 2. How the general fiber of the family in §3.2.2 moves.
and the non-zero restrictions of the generating codimension-two classes are thus
ψδ | =−2, ψδ | =−1, ψδ | =1, λδ | =−2,
0 S2 1,1 S2 2,1 S2 0 S2
λδ | =1, δ2| =−48, δ δ | =−10, δ δ | =12,
2,1 S2 0 S2 0 1,1 S2 0 2,1 S2
δ2 | =2, δ δ | =−1.
1,1 S2 1,1 2,1 S2
3.2.3. Let (E,p,q,r) be a general three-pointed elliptic curve. Consider the surface obtained by
identifying the point r with a moving point in E, and by attaching at the point q an elliptic tail
moving in a pencil of degree 12.
Figure 3. How the general fiber of the family in §3.2.3 moves.
The base of this surface is E×P1 =:S . The divisor classes restrict as follows
3
ψ| =π∗[p], λ| =π∗(x), δ | =−2π∗[r]+12π∗(x), δ | =−π∗(x).
S3 1 S3 2 0 S3 1 2 2,1 S3 2
We deduce that the non-zero restrictions of product of divisor classes are the following
ψλ| =1, ψδ | =12, ψδ | =−1, λδ | =−2, δ2| =−48, δ δ | =2.
S3 0 S3 2,1 S3 0 S3 0 S3 0 2,1 S3
Moreover,we have δ | =−π∗[q]·12π∗(x)=−12.
01a S3 1 2
3.3. Some equalities in A2(M ). In §4 we also use the following result.
3,1
LOCI OF CURVES WITH SUBCANONICAL POINTS IN LOW GENUS 9
Proposition 3.1. The following equalities hold in A2(M )
3,1
1 1
λδ = −ψ+ δ +δ δ ,
1,1 0 2,1 1,1
5(cid:18) 2 (cid:19)
δ = −12ψ2−24ψδ −372λ2+72λδ +120λδ −3δ2−12δ δ
00 1,1 0 2,1 0 0 2,1
−12δ2 −12δ2 +12κ ,
1,1 2,1 2
1 1 1 1 1
δ = ψ2+ ψδ +5λ2− λδ +4λδ − δ δ − δ δ
1|1 2 5 1,1 2 0 2,1 10 0 1,1 2 0 2,1
1 6 1 1 1
− δ2 − δ δ − δ2 + δ − κ ,
2 1,1 5 1,1 2,1 2 2,1 12 01a 2 2
78 55 3 7 17
γ = ψδ +126λ2− λδ −78λδ + δ2+ δ δ + δ δ
1 5 1,1 2 0 2,1 2 0 10 0 1,1 2 0 2,1
42
+12δ2 + δ δ +12δ2 ,
1,1 5 1,1 2,1 2,1
15 101
γ = ψ2−21ψλ+2ψδ +ψδ +3ψδ + λ2−10λδ −19λδ
2 0 1,1 2,1 0 2,1
2 2
1 1 5 1
+ δ2+2δ δ + δ2 + δ2 − κ .
2 0 0 2,1 2 1,1 2 2,1 2 2
Proof. The idea is to follow the lines of the proof of Proposition1.1. Let us start with the second
equality. Write δ as a linear combination of the classes in (1) with unknown coefficients. By
00
pulling-back via the map ϑ: M →M and using ϑ∗(δ )=δ and Mumford’s equality
2,2 3,1 00 00
δ =6λδ ∈R2(M ),
00 0 2,2
we are able to determine the coefficients in the statement up to three variables α,β,γ:
α+β
δ = (−2α−β)ψ2+6(α+β)ψλ− ψδ −2αψδ
00 0 1,1
2
α+β
+(−6α−5β+120γ)λ2+(6+ −22γ)λδ +(6α−4β)λδ
0 2,1
2
β−α
+γδ2+ δ δ −αδ2 +βδ2 ,+ακ ∈R2(M ).
0 2 0 2,1 1,1 2,1 2 3,1
Next,werestricttheaboveequalitytothethreetestsurfacesconsideredin§3.2. Itiseasytoshow
that the restriction of δ to the first test surface is 0, while is respectively
00
π∗(−2[r]−[q ])·π∗(12x)+π∗[q ]·π∗(12x)=−24
1 1 2 1 1 2
and
π∗(−2[r]−[p]−[q])·π∗(12x)+π∗[p]·π∗(12x)+π∗[q]·π∗(12x)=−24
1 2 1 2 1 2
to the other two test surfaces. It follows that α=12=−β and γ =−3, thus proving the equality
for δ .
00
The first equality is proven in a similar way. Alternatively, one obtains it as the push-forward
of divisor relations via the boundary map M ×M →M .
1,2 2,1 3,1
For the third equality, note that
ϑ∗(δ )=−ψ δ +δ ∈R2(M )
1|1 2 1,{1,2} 1|1 2,2
and by Getzler’s computations
1 7 1 7 1
δ = ψ2− ψ δ − ψ δ + ψ δ − ψ δ − ψ δ
1|1 2 2 1 1,{1} 10 2 1,{1} 2 1 1,{1,2} 10 2 1,{1,2} 10 2 0
1 1 3 11 1
+δ2 + δ δ + δ δ + δ δ + δ δ + δ2
0,{1,2} 5 1,{1,2} 0,{1,2} 10 0 0,{1,2} 50 0 1,{1} 600 0 1,{1,2} 200 0
10 DAWEICHENANDNICOLATARASCA
in R2(M ). Finally, the restriction of δ ∈R2(M ) to the third test surface in §3.2 is
2,2 1|1 3,1
π∗[r](−π∗(x))=−1,
1 2
while the restriction to the other two test surfaces is zero.
For γ and γ , we use
1 2
ϑ∗(γ )=γ +γ , ϑ∗(γ )=γ .
1 1 1,{1} 2 1,{2}
By Getzler’s computations, we have
6 9 6 24 1
γ =3ψ ψ +3ψ2− ψ δ − ψ δ − ψ δ − ψ δ − ψ δ
1 1 2 2 5 1 1,{1} 5 2 1,{1} 5 1 1,{1,2} 5 2 1,{1,2} 10 1 0
2 42 7 3 1
− ψ δ +12δ2 + δ δ + δ δ + δ (δ +δ )+ δ2,
5 2 0 0,{1,2} 5 1,{1,2} 0,{1,2} 10 0 0,{1,2} 25 0 1,{1} 1,{1,2} 100 0
6 33 6 18 1 3
γ =9ψ2−3ψ ψ + ψ δ − ψ δ + ψ δ − ψ δ + ψ δ − ψ δ ,
1,{1} 2 1 2 5 1 1,{1} 5 2 1,{1} 5 1 1,{1,2} 5 2 1,{1,2} 10 1 0 10 2 0
24 3 36 48 3 4
γ =9ψ2−3ψ ψ − ψ δ − ψ δ + ψ δ − ψ δ + ψ δ − ψ δ
1,{2} 2 1 2 5 1 1,{1} 5 2 1,{1} 5 1 1,{1,2} 5 2 1,{1,2} 5 1 0 5 2 0
in R2(M ). The intersection of γ ∈R2(M ) with the first test surface in §3.2 is
2,2 1 3,1
(−π∗([p]+[q ])−π∗[q ])·(−π∗[q ]−π∗[q ])+π∗[p](−π∗[q ])=2,
1 1 2 2 1 1 2 2 1 2 2
while it is zero with the other two test surfaces. Similarly, the intersection of γ ∈R2(M ) with
2 3,1
the first test surface in §3.2 is
π∗[p](−π∗[q ])=−1,
1 2 2
while it is zero with the other two test surfaces. (cid:3)
3.4. A check. As a partial check, we compute the expressions for δ and γ in an alternative
00 1
way. The classes δ and γ in R2(M ) coincide with the pull-back via p: M → M of the
00 1 3,1 3,1 3
classes δ and γ in A2(M ). Using the following identities in A2(M )
00 1 3 3
δ = −372λ2+72λδ +120λδ −3δ2−12δ δ −12δ2+12κ ,
00 0 1 0 0 1 1 2
55 3 17
γ = 126λ2− λδ −78λδ + δ2+ δ δ +12δ2
1 2 0 1 2 0 2 0 1 1
from [Fab90a, Theorem 2.10], and the following well-known identities
p∗(λ)=λ, p∗(δ )=δ , p∗(δ )=δ +δ , κ =p∗(κ )+ψ2,
0 0 1 1,1 2,1 2 2
one recovers the two formulae in Proposition 3.1.
4. The locus of Weierstrass points on hyperelliptic curves in M
3,1
Inthis sectionwecompute the classofthe closurein M ofthe locusofWeierstrasspoints on
3,1
hyperelliptic curves of genus 3
Hyp :={[C,p]∈M |C is hyperelliptic and p is a Weierstrass point in C}.
3,1 3,1
The idea is to consider the pull-back of the hyperelliptic locus in M via the clutching map
4
j : M → M obtained by attaching a fixed elliptic tail at the marked point of an element
3 3,1 4
[C,p]∈M . Using the theory of admissible covers,the following statement is straightforward.
3,1
Lemma 4.1. We have the following equality in A2(M )
3,1
j∗ Hyp = Hyp .
3 4 3,1
(cid:0) (cid:1)