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.

[CT] A CASTRAVET AND E TEVELEV. Hypertrees, Projections, and Moduli of Stable RationalCurves. CRELLE’S JOURNAL, 675 (2013), 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.

[S] I SCHEIDWASSER.Hypergraph Curves. HONORS THESIS AT THE UNIVERSITY OF MAS-SACHUSETTS AMHERST. WWW.MATH.UMASS.EDU/˜TEVELEV.