EFFECTIVE DIVISORS ON MODULI SPACES OF CURVES WITH MARKED
POINTSMorgan Opie
University of Massachusetts Amherst
1 IntroductionWe study effective divisors on the moduli space
M0,n of stable rational curves with markedpoints, focusing on
hypertree divisors and Chen-Coskun divisors introduced in [CT] and
[CC],respectively. After preliminaries onM0,n, we define hypertree
divisors as studied by Castravet-Tevelev, and discuss our database
of hypertree divisor classes. We then introduce Chen-Coskun
divisors on M1,n, and relate new results regarding their proper
transforms on M0,n+2.From this we construct an extremal divisor on
M0,7 outside the effective span of hypertree andboundary classes,
providing a counterexample to a conjecture from [CT].
2 Background: M0,nThe moduli space M0,n parametrizes equivalence
classes of n marked points on P1 underthe action of PGL2. Its
compactification, M0,n, parametrizes nodal trees of P1’s with n
mark-ings and with each component P1 having at least 3 “special”
points (marked points or nodes),modulo automorphisms.
EXAMPLES OF STABLE RATIONAL CURVES: N=5.
For I ⊂ {1, ..., n} with 1 < |I|, |Ic|, a boundary divisor δI
is the collection of stable rationalcurves in M0,n with a node
separating the marked points in I and Ic.
3 Divisor classes on M0,nThe following diagram is useful in
studying divisors on M0,n:
An+1 α←− A1[n + 1] β−→ M0,n+1π−→ M0,n
ψ
yPn−2
Above, α is an iterated blow-up of An+1 along partial diagonals
defining Fulton-Macphersonconfiguration space A1[n + 1], which is a
partial compactification of the space of n + 1 distinctmarked
points in A1 [FM]. This gives a basis of Cl(A1[n + 1]): exceptional
divisors ∆I overdiagonals {xi = xj : i, j ∈ I} for 2 ≤ |I| ≤ n − 1.
On the interior, β takes a marked A1 toits isomorphism class in
M0,n+1. The map π “forgets” the n + 1 marking on a given
stablerational curve. The map ψ is the Kapranov morphism: fix p1,
..., pn ∈ Pn−2. For I ⊂ {1, ..., n},ψ(δI∪{n+1}) is the hyperplane
spanned by pi for i ∈ I [K]. This gives a set of free genera-tors
for Cl(M0,n+1): H = ψ−1∗ (h), for h ⊂ Pn−2 a hyperplane, together
with EI = δI,n+1 forI ⊂ {1, ..., n} satisfying 1 < |I| < n−
2.We can specify a prime, non-boundary divisor D ⊂ M0,n by an
irreducible polynomialf (x1, ..xn) such that [P1;x1, . . . , xn] ∈
D ∩ M0,n iff (x1, .., xn) ∈ V (f ) ⊂ An. D ⊂ M0,n isthe closure of
D ∩M0,n, and α−1∗ (V (f )) = β−1∗ (D). The class of α−1∗ (D) is
computable frommultiplicities of f along partial diagonals, which
gives the followingTheorem. Given an irreducible polynomial f ∈
k[x1, .., xn] specifying a divisor in M0,n, theclass of the proper
transform in M0,n+1 under the forgetful morphism in index n + 1
isdH −
∑mIEI , where mI is the multiplicity of f along the partial
diagonal {xi = xj : i, j /∈ I},
and d = deg(f ).
With the above result, we wrote a Macaulay 2 program to compute
classes of divisors on M0,nspecified by irreducible polynomial
equations. Applications of this program are discussed insection
5.
4 Hypertree divisorsThe following definitions are those of
Castravet-Tevelev.A hypertree on a set N with is a collection Γ =
{Γ1, . . . ,Γd} of subsets of N satisfying:
• ∀j, |Γj| ≥ 3• i ∈ N =⇒ ∃j, k ≤ d, j 6= k with i ∈ Γj and i ∈
Γk• convexity:
∣∣⋃j∈S Γj
∣∣−2 ≥∑j∈S(|Γj| − 2), ∀S ⊂ {1, . . . , d}• normalization: |N | −
2 =
∑j≤d(|Γj| − 2)
Γ is irreducible if the convexity condition is strict for 1 <
|S| < d. Let |N | = n. All hypertreedivisors for n ≤ 11 were
found in [S]. Enumeration of irreducible hypertrees is as follows:
onefor n = 6, n = 7; three for n = 8; eleven for n = 9; and 96 for
n = 10. A planar realizationp1, ..., pn ∈ P2 for a hypertree Γ
satisfies pi, pj, pk are collinear⇐⇒ ∃α such that i, j, k ∈ Γα.
PLANAR REALIZATION FOR N=6, Γ = {012, 314, 045, 325}.
A hypertree divisor DΓ ⊂M0,n is the closure of the locus{[P1;
q1, ..., qn] : ∃ a realization {pi} of Γ and projection π with qi =
π(pi)
}⊂M0,n.
For Γ irreducible, Castravet-Tevelev show DΓ is a nonempty
irreducible divisor generating anextremal ray of Eff (M0,n).
Castravet-Tevelev also state the followingConjecture. The effective
cone of M0,n is generated by boundary divisors and by divisors
DΓparametrized by irreducible hypertrees and their pullbacks.
A counterexample to this conjecture is provided in section
7.
5 Hypertree resultsWe obtained equations in k[x1, ..., xn]
specifying irreducible hypertree divisors, generaliz-ing results in
[CT] for the case |Γi| = 3, ∀i. Using our Macaulay program for
com-puting classes specified by polynomial equations, we computed
all divisor classes corre-sponding to irreducible hypertrees for 6
≤ n ≤ 10. We additionally wrote a programto compute symmetry group
sizes, and computed symmetry groups of irreducible hyper-tree
classes for 6 ≤ n ≤ 9. Particularly nice “spherical” hypertrees are
obtained viapolygonal tilings of a two-sphere, see [CT]; for 6 ≤ n
≤ 10, these were classified.A complete database with this
information, along with Macaulay code, can be found
athttps://www.math.umass.edu/~tevelev/HT_database/database.html. It
is hoped that thesedata will prove useful in further investigations
of hypertrees and other divisors. A taste of thedatabase is
provided below.Example. The only irreducible hypertree for n=7 is Γ
= {012, 034, 135, 056, 246}.DΓ =
−E1234−E0135−E1235−E1345−E0246−E1246−E2346−E1256−E1356−E2456−E3456−E012−E013−E123−E024−E124−E034−E134−E234−E015−E125−E035−2E135−E235−E145−E245−E345−E026−E126−E136−E236−E046−E146−2E246−E346−E056−E156−E256−E356−E456−E01−E02−2E12−E03−2E13−2E23−E04−2E14−2E24−2E34−E05−2E15−2E25−2E35−2E45−E06−
2E16− 2E26− 2E36− 2E46− 2E56− 2E0− 3E1− 3E2− 3E3− 3E4− 3E5− 3E6 +
4H.Symmetry group order 12.Example. A sampling of spherical
hypertree sketches:
{012, 314, 045, 325} {0123, 456, 047, 158, 267, 348} {012, 345,
036, 146, 057, 247}{012, 345, 067, 138, 469, 258, 039, 247} {012,
345, 067, 138, 049, 236, 058, 379}
6 Chen-Coskun divisorsM1,n is the moduli space of stable genus 1
curves with n ordered marked points; a generalpoint is of the form
[C; p1, ..., pn] for C a smooth genus 1 curve and pi ∈ C marked
points.Chen-Coskun define divisors on M1,n as follows: for a = (a1,
..., an) satisfying
∑i ai = 0, de-
fine Da to be the closure of smooth [C; p1, ..., pn]
satisfying∑aipi = 0 ∈ Cl(C). A result of
Chen-Coskun is that for n ≥ 3, gcd(a1, ..., an) = 1, Da is an
extremal and rigid effective divisor.We have a map from M0,n+2 to
M1,n which identifies marked points pn+1 and pn+2:
Hence it is natural to consider proper the transform of Da on
M0,n+2, which we will denoteD̃a and refer to as a CC divisor. That
D̃a is effective follows from definitions, but it is easy to
construct examples of non-extremal CC divisors (e.g. any CC
divisor on M0,5). We presentresults on CC divisors, with an eye
towards the question of which are extremal.Theorem. For I ⊂ {1,
..., n} and D̃(a1,...,an) ⊂ M0,n+2, the class of the proper
transform inM0,n+3 with respect to the forgetful morphism is dH
−
∑mIEI , where mI , d are as follows:
• n + 1, n + 2 /∈ I : mI = (∑i/∈I |ai|)− 1.
• n + 1, n + 2 ∈ I : mI = 0 unless I = {n + 1, n + 2} in which
case mI = 1.• |{n + 1, n + 2} ∩ I| = 1 : mI = min{
∑0≤ai/∈I |ai|,
∑0≥ai/∈I |ai|}.
• d = (∑i |ai|)− 1.
Proof. The argument consists of deriving rational equations for
CC divisors, and applying thetheorem from section 3. The equation
is
1
pn+1 − pn+2
( ∏ai≥0
(pn+1−pi)|ai|∏ai≤0
(pn+2−pi)|ai| −∏ai≤0
(pn+1−pi)|ai|∏ai≥0
(pn+2−pi)|ai|)
The next results relate hypertree divisors to CC
divisors:Theorem. If a = (1, 1, . . . ,−1,−1, . . .), D̃a = DΓ
where Γ is the spherical hypertree divisorassociated to a bipyramid
triangulation.Theorem. Given D̃(a1,...,an,an+1) ⊂M0,n+4 with an,
an+1 ≥ 0, D̃a∩ δn,n+1 ⊂ δn,n+1 'M0,n+3 hasclass of D̃α, where α =
(a1, ..., an + an+1).
Together, the previous two theorems shows that all CC divisors
are obtained as iterated re-strictions of bipyramid hypertree
divisors.
7 CounterexampleLet a=(2,1,-1,-1,-1). Using 1 as our special
index, define a Kapranov map from M0,7 to P4. Thecorresponding D̃a
⊂M0,7 has class 3H−2E1−2E2−2E3−2E4−E5−E6−E236−E235−E345−E346 − E246
− E245 − . . ., and a Macaulay computation shows that the image of
D̃a under thisKapranov map is a cubic threefold with singularities
at most nodal, and in particular a node atp1. To show D̃a is
extremal, it suffices to exhibit a family irreducible curves C
covering D̃a, withC · D̃a < 0.
Consider the family of curves obtained byintersecting a 2-plane
through p1 with S =ψ(D̃a). For irreducibility, one uses
Bertini’stheorem. A general curve from the irre-ducible covering
family on M0,7 obtainedby taking “proper transforms” of the
cover-ing family for S has intersection pairing −1with D̃a: its
image in P4 passes throughp1 with multiplicity two, intersects a
gen-eral 2-plane (in particular, the six 2-planeswhose proper
transforms contribute to theclass of D̃a) transversally, and
intersects ahyperplane in three points.
This gives 3 ∗ 3 − 2 ∗ 2 − 1 − 1 − 1 − 1 − 1 − 1 = −1.
Inspection shows that the class of D̃a isnot a hypertree divisor
class.
8 Future plansWe are working towards a proof that CC divisors
for a of the form (n, 1,−1,−1, ...) are ex-tremal. Future
investigation will likely focus on extremality of other interesting
CC divisors, andon connections between CC divisors and hypertree
divisors.
9 References[CC] D CHEN AND I COSKUN.Extremal Effective Divisors
on M1,n. ARXIV: 1304.0305V1.
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121-180.
[FM] W FULTON AND R MACPHERSON. A Compacitification of
Configuration Spaces. ANNALSOF MATHEMATICS, 1994.
[K] M KAPRANOV. Chow Quotients of Grassmanians I. I. M. GELFAND
SEMINAR, ADV. SO-VIET MATH. 16, PART 2, AMER. MATH. SOC.,
PROVIDENCE, 1993, 29110.
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