On S n -invariant conformal blocks vector bundles of rank one on M ¯ ¯ ¯ ¯ ¯ 0,n
aa r X i v : . [ m a t h . AG ] A p r ON S n -INVARIANT CONFORMAL BLOCKS VECTOR BUNDLES OFRANK ONE ON M , n ANNA KAZANOVA
Abstract.
For any simple Lie algebra, a positive integer, and tuple of compatible weights,the conformal blocks bundle is a globally generated vector bundle on the moduli space ofpointed rational curves. We classify all S n -invariant vector bundles of conformal blocks for sl n which have rank one. We show that the cone generated by their base point free firstChern classes is polyhedral, generated by level one divisors. Introduction
To any simple Lie algebra g , positive integer ℓ , and n -tuple ~λ , of dominant weights for g at level ℓ , there is a globally generated vector bundle V ( g , ~λ, ℓ ) of conformal blocks on themoduli space M ,n , of stable n -pointed rational curves [TUY89, Fak12]. Their first Chernclasses, the conformal blocks divisors D ( g , ~λ, ℓ ), are base point free, and therefore lie in thecone of nef divisors.Understanding the nef divisors on a variety is central to understanding its birational ge-ometry. Vector bundles of conformal blocks and their Chern classes have been studied,primarily with standard intersection-theoretic methods, using Fakhruddin’s formulas for theChern classes and their intersections with F-curves [AGSS11, Fed11, Swi11, BG12, Fak12,GG12, Fed13, Gia13, GJMS13, AGS14]. Examples of conformal blocks divisors can be com-puted using Swinarski’s implementation of these formulas into Macaulay2 software, [Swi10].While recursive, and dependent on the computation of ranks of the bundles, computationsare limited to divisors of relatively low level on M ,n for low n . Many open questions aboutthe divisors persist.In this paper we study the subcone of the nef cone generated by an infinite set of divisors S , consisting of the first Chern classes of conformal blocks vector bundles of rank 1 for sl n with S n -invariant weights. This is a generalization of [AGSS11], where the authors studieda set of ⌊ n/ ⌋ S n -invariant sl n divisors of level one, which are all first Chern classes of rankone bundles. In that paper, it was shown that each level one divisor spanned an extremal rayof the S n -invariant nef cone of M ,n . While our family consists of infinitely many divisors,we prove that they all are contained in the cone generated by the original divisors studiedin [AGSS11]. This work can be seen as an illustration for how to study nontrivial families of conformalblock divisors on M , n using Schubert calculus and tools from [BGM13].For the finite dimensional simple Lie algebra sl n , the dominant integral weights λ areparameterized by Young diagrams λ = ( λ (1) ≥ λ (2) ≥ · · · ≥ λ ( n − ≥ λ ( n ) ≥
0) with λ (1) ≤ ℓ .We use the notation | λ | = P ni =1 λ ( i ) . As is standard, we denote the fundamental dominantweights of the form λ = (1 , . . . , , , . . . ,
0) with | λ | = i by ω i .Our main theorem provides a complete description of S n -invariant rank 1 vector bundlesfor sl n on M ,n . Theorem 1.1.
Let
Λ = { ( ℓ − m ) ω i + mω i +1 : 0 ≤ m ≤ ℓ, ≤ i ≤ n − } . Then we have (1) rk V ( sl n , λ n , ℓ ) = 1 if λ ∈ Λ . (2) rk V ( sl n , λ n , ℓ ) > if λ Λ . For any fixed n > S to be the set of of all Chern classes of rank 1 vectorbundles, as classified by the Theorem 1.1: S = { c V ( sl n , (( ℓ − m ) ω i + mω i +1 ) n , ℓ ) : ℓ > , ≤ m ≤ ℓ, ≤ i ≤ n − } . There are tools for studying rank one bundles (see Section 2), which we use to give thefollowing simple description as positive linear combinations of level one divisors.
Proposition 1.2.
For any D ∈ S we have the following decomposition: D ( sl n , (( ℓ − m ) ω i + mω i +1 ) n , ℓ ) = ( ℓ − m ) D ( sl n , ( ω i ) n ,
1) + m D ( sl n , ( ω i +1 ) n , . It therefore follows that the cone generated by this infinite set of divisors S is in factpolyhedral: Corollary 1.3.
The cone of divisors generated by S is the convex hull of the ⌊ n ⌋ − extremalrays of the S n -invariant nef cone spanned by divisors D ( sl n , ω ni , , where ≤ i ≤ ⌊ n ⌋ . Acknowledgements.
I am grateful to Angela Gibney for many useful discussions, com-ments, and encouragement. I thank Prakash Belkale for pointing out the proof of theLemma 4.2, and for the comments on a draft of the paper. I also thank Dustin Cartwrightand Linda Chen for helpful conversations.2.
Tools for computing ranks of conformal blocks bundles
We refer the reader to [BGM13] for background information on conformal blocks vectorbundles and divisors. To compute the ranks of the vector bundles of conformal blocks, weuse a special case of “Witten’s Dictionary”, which is covered in Section 2.1. The classicaland quantum versions of the Pieri and Giambelli formulas are as often useful for applyingWitten’s dictionary, and we state those in Section 2.2.
N S n -INVARIANT CONFORMAL BLOCKS VECTOR BUNDLES OF RANK ONE ON M , n Cohomological version of Witten’s Dictionary.
Recall that the (small) quan-tum cohomology ring QH ∗ (Gr( n, E ); Z ), for E ∼ = C n + ℓ , is a Z [ q ]-algebra isomorphic toH ∗ (Gr( n, E ); Z ) ⊗ Z Z [ q ], as a module over Z [ q ], with elements σ λ = σ λ ⊗
1, where σ λ ∈ H ∗ (Gr( n, E ); Z ) the cohomology class corresponding to the Schubert variety Ω λ ( F • ), where F • is a full flag and λ a partion (see for example [Ber97] for the definitions).To compute the rank of our particular S n -invariant conformal blocks bundles V ( sl n , λ n , ℓ ),for sl n , one proceeds as follows. For | λ | = k , write k = ℓ + s . There are two cases:(1) If s ≤
0, then rk V ( sl n , ~λ, ℓ ) is equal to the coefficient of the class of a point σ kω n inthe product σ λ · · · σ λ n ∈ H ∗ (Gr( n, C n + k )) . (2) If s >
0, then rk V ( sl n , ~λ, ℓ ) is equal to the coefficient of the class of q s [pt] , where([pt] = σ ℓω n = σ ( ℓ,...,ℓ ) ) in the product σ λ ⋆ · · · ⋆ σ λ n ⋆ σ sℓω ∈ QH ∗ (Gr( n, E )) . See [Bel08, Theorem 3.6 and Remark 3.8] for the most general statement and proof of Witten’sDictionary.2.2.
Pieri and Giambelli formulas.
For convenience, we state the classical and quantumversions of the Pieri and Giambelli formulas here.
Classical Pieri formula.
If the Young diagram associated to λ is contained in a n × ℓ grid,and i ≤ n , then the product of Schubert classes in the cohomology ring H ∗ Gr( n, ℓ + n ) isgiven by σ λ · σ pω = X σ π , where the sum is over all partitions π obtained by adding i boxes to λ , no two in the samecolumn. Quantum Pieri formula. [Ber97] If the Young diagram associated to λ is contained in a n × ℓ grid, and p ≤ ℓ , then the product of Schubert classes in the quantum cohomology ringQH ∗ Gr( n, ℓ + n ) is given by σ λ ⋆ σ pω = X σ µ + q X σ ν , where the first sum is over all partitions µ obtained by adding p boxes to λ , no two in thesame column, and the second sum is over all partitions ν obtained by removing n + ℓ − p boxes from λ , at least one from each column. Quantum Giambelli. [Ber97] Set σ = 1, σ i = 0 for i < i > ℓ .If λ is a partition contained in an n × ℓ rectangle, then the Schubert class σ λ ∈ QH ∗ Gr( n, n + ℓ ) is given by σ λ = det( σ λ ( i ) + j − i ) ≤ i,j ≤ n . ANNA KAZANOVA Rank one bundles
In this section we prove the first part of Theorem 1.1, which states that the bundles V ( sl n , ( λ ) n , ℓ ) have rank one if λ ∈ Λ. We will need the following Lemma.
Lemma 3.1.
Let λ = ( λ (1) , . . . , λ ( n − , be such that λ (1) = · · · = λ ( i ) = ℓ , λ ( i +1) < ℓ .Denote by µ the partition ( λ ( i +1) , . . . , λ ( n − , , . . . , . Then (3.1) rk V ( sl n , λ n , ℓ ) = rk V ( sl n , µ n , ℓ ) . Proof.
Using Giambelli formula, we can write λ = σ iℓ µ since λ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ ℓ . . . . . . ∗ σ ℓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ ∗ . . . σ ℓ . . . ∗ ∗ . . . ∗ σ λ ( i +1) . . . ∗ ∗ . . . . . . . . . . . . . . . . . . . . . . . . ∗ ∗ . . . ∗ ∗ . . . σ λ ( n − ∗ . . . . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = σ iℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ λ ( i +1) . . . ∗ ∗ ∗ . . . ∗ . . . . . . . . . . . . . . . . . . . . . ∗ . . . σ λ ( n − ∗ ∗ . . . ∗ . . . ∗ . . . ∗ . . . . . . ∗ . . . . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = σ iℓ µ. We compute rk V ( sl n , λ n , ℓ ) using Witten’s dictionary. Write n | λ | = nℓ + ns , so that s = | λ | − ℓ , and then rk V ( sl n , λ n , ℓ ) is equal to a coefficient of q s [pt] = q s σ ( ℓ,...,ℓ ) in thequantum product σ ⋆nλ ⋆ σ sℓ ∈ QH ∗ (Gr( n, n + ℓ ))) . We have σ ⋆nλ ⋆ σ sℓ = σ ⋆inℓ ⋆ σ ⋆nµ ⋆ σ sℓ . Write | λ | = iℓ + | µ | , then s = ( i − ℓ + | µ | , so that σ ⋆nλ ⋆ σ sℓ = σ ⋆inℓ ⋆ σ ⋆nµ ⋆ σ ⋆ (( i − ℓ + | µ | ) ℓ = σ ⋆ ( in +( i − ℓ + | µ | ) ℓ ⋆ σ ⋆nµ . Note that σ ⋆nℓ = σ ( ℓ,...,ℓ ) and σ ⋆ ( n + ℓ ) ℓ = q ℓ σ (0 ,..., by quantum Pieri rule.Thus, we have σ ⋆ ( i − ℓ + n ) ℓ = q ( i − ℓ σ (0 ,..., . Therefore σ ⋆nλ ⋆ σ sℓ = q ( i − ℓ σ ⋆ ( n + | µ | ) ℓ ⋆ σ ⋆nµ . If | µ | > ℓ , then σ ⋆nλ ⋆ σ sℓ = q iℓ σ ⋆ ( | µ |− ℓ ) ℓ ⋆ σ ⋆nµ . In this case the coefficient of q s [pt] in thequantum product σ ⋆nλ ⋆ σ sℓ is equal to the coefficient of q | µ |− ℓ [pt] in the quantum product σ ⋆ ( | µ |− ℓ ) ℓ ⋆ σ ⋆nµ ∈ QH ∗ (Gr( n, n + ℓ ))) . Note that the latter equals to the rk V ( sl n , µ n , ℓ ) byWitten’s dictionary.If | µ | ≤ ℓ , then σ ⋆nλ ⋆ σ sℓ = q i ( ℓ − σ ⋆ ( n + | µ | ) ℓ ⋆ σ ⋆nµ . Since by the Pieri rule, σ ⋆ ( n + | µ | ) ℓ = q | µ | σ ( ℓ −| µ | ,...,ℓ −| µ | ) , we conclude that σ ⋆nλ ⋆ σ sℓ = q i ( ℓ − | µ | σ ( ℓ −| µ | ,...,ℓ −| µ | ) ⋆ σ ⋆nµ = q s σ ( ℓ −| µ | ,...,ℓ −| µ | ) ⋆σ ⋆nµ . In this case the coefficient of q s [pt] in the quantum product σ ⋆nλ ⋆ σ sℓ is equal to the co-efficient of σ ( ℓ,...,ℓ ) in the classical product σ ( ℓ −| µ | ,...,ℓ −| µ | ) · σ nµ ∈ H ∗ Gr( n, n + ℓ ). Note that thiscoefficient is equal to the coefficient of σ ( | µ | ,..., | µ | ) in the classical product σ nµ ∈ H ∗ Gr( n, n + | µ | ),which is equal to rk V ( sl n , µ n , ℓ ) by Witten’s dictionary. (cid:3) Proposition 3.2. V ( sl n , (( ℓ − m ) ω i + mω i +1 ) n , ℓ ) are rank one bundles.Proof. By Proposition 3.1, we have rk V ( sl n , (( ℓ − m ) ω i + mω i +1 ) n , ℓ ) = rk V ( sl n , ( mω ) n , ℓ ). N S n -INVARIANT CONFORMAL BLOCKS VECTOR BUNDLES OF RANK ONE ON M , n Since n · | mω | = nm ≤ nℓ , by Witten’s dictionary, rk V ( sl n , ( mω ) n , ℓ ) is equal to themultiplicity of class of the point σ ( m,...,m ) in the product σ nm ∈ H ∗ Gr( n, n + m ). Since σ nm = σ ( m,...,m ) by Pieri rule, we have rk V ( sl n , ( mω ) n , ℓ ) = 1, and we conclude that the rankrk V ( sl n , (( ℓ − m ) ω i + mω i +1 ) n , ℓ ) is equal to 1. (cid:3) Higher rank bundles
Recall from the Introduction that Λ is the set of weights { ( ℓ − m ) ω i + mω i +1 : 0 ≤ m ≤ ℓ, ≤ i ≤ n − } . In Proposition 4.3, we will show that rk V ( sl n , µ n , ℓ ) >
1, for µ Λ.Combined with Proposition 3.2, this completes the proof of Theorem 1.1. We begin withsome special cases, and the main proof will reduce to the special cases.
Lemma 4.1.
We have rk V ( sl n , ( ω i ) n , > for < i < n − and n ≥ .Proof. Since rk V ( sl n , ( ω i ) n ,
2) = rk V ( sl n , ( ω n − i ) n , i ≤ ⌊ n/ ⌋ .Suppose that i = 2. By Witten’s dictionary, we have to find the coefficient of σ (2 ,..., inthe classical product σ nω . Note that σ ω = σ ω + σ ω + ω + σ ω . For each of these terms wehave σ n − ω · σ ω = a σ (2 ,..., , σ n − ω · σ ω + ω = a σ (2 ,..., , and σ n − ω · σ ω = a σ (2 ,..., , where eachof the constants a , a , and a are at least 1. Thus rk V ( sl n , ( ω ) n , ≥ i >
2. Since ni = 2 n + ( i − n , by Witten’s dictionary, we need tocompute the coefficient of qσ (2 ,..., in the quantum product σ ⋆nω i ⋆ σ i − ω .Write n = iq + r , where 0 ≤ r < i , let α = q +1. Then σ ⋆nω i ⋆σ i − ω = σ n − α ( i − ω i ⋆σ ⋆α ( i − ω i ⋆σ i − ω ,and σ ⋆αω i ⋆σ ω = cσ ω n + ω i − r ⋆σ ω +other terms = cqσ ω r − i +other terms . Note that since α ≥ c ≥
2, and all the other terms have nonnegative coefficients, since a productof effective cycles is effective. Thus σ ⋆nω i ⋆ σ i − ω = c i − q i − σ n − α ( i − ω i ⋆ σ i − ω r − i + other terms . Since σ n − α ( i − ω i ⋆ σ i − ω r − i = σ (2 ,..., + other terms, we conclude that rk V ( sl n , ( ω i ) n , ≥ c i − ,and in particular, we have rk V ( sl n , ( ω i ) n , > (cid:3) Lemma 4.2.
We have rk V ( sl n , (( aω i + bω i +1 ) n , a + b + 1) > for all a ≥ , b > , < i 1) = 1. Thereforeby [BGM13, Prop. 17.1], the map V ( sl n , ( aω i + bω i +1 ) n , a + b + 1) → V ( sl n , ( aω i + ( b − ω i +1 ) n , a + b − ⊗ V ( sl n , ( ω i +1 ) n , V ( sl n , ( aω i + bω i +1 ) n , a + b + 1) is greater than or equal tothe rank rk V ( sl n , ( ω i +1 ) n , V ( sl n , ( ω i +1 ) n , > (cid:3) Proposition 4.3. We have rk V ( sl n , µ n , ℓ ) > , if µ Λ . ANNA KAZANOVA Proof. Throughout the proof we will use the following classical formula (see for exam-ple [BGM13, Lemma 1.8]). For all c > 0, we have(4.1) rk V ( sl n , µ n , ℓ ) ≤ rk V ( sl n , µ n , ℓ + c ) . (1) First, suppose that µ = bω i . Then since µ Λ, we have 1 < i < n − 1, and b < ℓ . In this case, rk V ( sl n , µ n , ℓ ) ≥ rk V ( sl n , ( bω i ) n , b + 1) by (4.1). We haverk V ( sl n , ( bω i ) n , b + 1) > µ = aω i + bω j , where i < j , a, b = 0 and ℓ ≥ ( a + b ).(i) If ℓ = a + b , then since µ Λ, we have j ≥ i + 2. By Theorem 3.1,rk V ( sl n , µ n , ℓ ) = rk V ( sl n , ( bω j ) n , a + b ). Now we reduced the problem to thecase (1).(ii) If ℓ > a + b , and j > i + 1, we can apply (4.1) to conclude that rk V ( sl n , µ n , ℓ ) ≥ rk V ( sl n , ( aω i + bω j ) n , a + b ), so we reduced to the case (2i).(iii) If ℓ > a + b , and j = i + 1, we can apply (4.1) to conclude that rk V ( sl n , µ n , ℓ ) ≥ rk V ( sl n , ( aω i + bω i +1 ) n , a + b + 1). We have rk V ( sl n , ( aω i + bω i +1 ) n ,a + b +1) > i < · · · < i k be an ordered subset of { , . . . , n − } , and let µ = P kj =1 c i j ω i j with all c i j > 0, and ℓ ( µ ) = P kj =1 c i j ≤ ℓ , and k ≥ 3. By (4.1) we haverk V ( sl n , µ n , ℓ ) ≥ rk V ( sl n , µ n , ℓ ( µ )) . Using Proposition 3.1, we obtain thatrk V ( sl n , µ n , ℓ ( µ )) = rk V ( sl n , ( k X j =2 c i j ω i j ) n , ℓ ( µ )) , so that rk V ( sl n , µ n , ℓ ) ≥ rk V ( sl n , ( k X j =2 c i j ω i j ) n , ℓ ( µ )) . We can repeat this process k − V ( sl n , µ n , ℓ ) ≥ rk V ( sl n , ( c i k − ω i k − + c i k ω i k ) n , c i k − + c i k − + c i k ) . Thus we reduced to the case (2ii) if i k > i k − + 1, and the case (2iii) if i k = i k − + 1. (cid:3) Example 4.4. In this example we elaborate on the computation in the proof of Lemma 4.1 toshow that rk V ( sl , ω , ≥ . Using Witten’s dictionary, we need to compute the coefficientof qσ (2 , , , , , , in the quantum product σ ⋆ ω ⋆ σ ω .We see that α = 3 , and N S n -INVARIANT CONFORMAL BLOCKS VECTOR BUNDLES OF RANK ONE ON M , n σ ⋆ ω = 3 + other effective terms . Then by Pieri rule, σ ⋆ ω ⋆ σ ω = 3 q + other effective terms . Finally, σ ⋆ ω ⋆ σ ω =3 q + other effective terms . In particular, rk V ( sl , ω , ≥ . Decomposition of D ( sl n , (( ℓ − m ) ω i + mω i +1 ) n , ℓ )In this section we identify all D ∈ S as effective sums of level one divisors, and from thisconclude that the cone S is finitely generated. Proof of Proposition 1.2. By Proposition 3.2, we have rk V ( sl n , ( ℓ − m ) ω i + mω i +1 , ℓ ) = 1.Moreover, since we know that the level one bundles V ( sl n , ω nj , 1) have rank one, and so by Bel-kale’s quantum generalization of Fulton’s conjecture [Bel07], rk V ( sl n , ( N ω j ) n , N ) = 1 for allintegers N . We have that rk V ( sl n , (( ℓ − m ) ω i ) n , ( ℓ − m )) = 1 and rk V ( sl n , ( mω i +1 ) n , m ) = 1.So by applying [BGM13, Prop. 17.1] and [BGM13, Cor. 17.3], we conclude that D ( sl n , (( ℓ − m ) ω i + mω i +1 ) n , ℓ ) = D ( sl n , ( ℓ − m ) ω ni , ℓ − m ) + D ( sl n , mω ni +1 , m )= ( ℓ − m ) D ( sl n , ω ni , 1) + m D ( sl n , ω ni +1 , . (cid:3) Remark 5.1. Without loss of generality assume that m > . Using Macaulay2, we checkedthat up to n = 2000 , each family F nℓ,m = { D ( sl n , (( ℓ − m ) ω i + mω i +1 ) n , ℓ ) : 1 ≤ i ≤ ⌊ n/ ⌋ − } gives a basis of Pic(M ,n ) S n by intersecting divisors from the family F nℓ,m with the independentset { F , ,i,n − i − } ⌊ n/ ⌋− i =1 of F–curves on (M ,n ) S n using [Fak12, Proposition 5.2] . We checked the full dimensionality of S for all n ≤ n , and we believe that the statement of Remark 5.1 is true for all n .So at least up to n = 2000, the cone generated by S is full dimensional as it contains allthe full dimensional cones generated by the F nℓ,m . By Proposition 1.2, each divisor D ∈ S isa linear combination of D ( sl n , ω ni , ≤ i ≤ ⌊ n/ ⌋ , by [AGSS11] define extremalrays of the S n -invariant nef cone Nef(M ,n ) S n . So the cone generated by S is equal to thecone spanned by these rays. ANNA KAZANOVA References [AGS14] Valery Alexeev, Angela Gibney, and David Swinarski, Higher level conformal blocks on M ,n from sl , Proc. Edinb. Math. Soc., (2014), 7-30.[AGSS11] Maxim Arap, Angela Gibney, Jim Stankewicz, and David Swinarski, sl n level Conformal blocksdivisors on M ,n , International Math Research Notices (2011).[Bel07] Prakash Belkale, Geometric proof of a conjecture of Fulton , Adv. Math. (2007), no. 1, 346–357.[Bel08] , Quantum generalization of the Horn conjecture , J. Amer. Math. Soc. (2008), no. 2,365–408.[BGM13] Prakash Belkale, Angela Gibney, and Swarnava Mukhopadhyay, Quantum cohomology and con-formal blocks on M ,n (2013). arXiv:1308.4906v3 [math.AG].[Ber97] Aaron Bertram, Quantum Schubert Calculus , Adv. Math (1997), 289–305.[BG12] Michelle Bolognesi and Noah Giansiracusa, Factorization of point configurations, cyclic covers,and conformal blocks , J. Eur. Math. Soc, to appear, (2012). arXiv:1208.4019 [math.AG].[Fak12] Najmuddin Fakhruddin, Chern classes of conformal blocks , Compact moduli spaces and vectorbundles, Contemp. Math., vol. 564, Amer. Math. Soc., Providence, RI, 2012, pp. 145–176.[Fed11] Maksym Fedorchuk, Cyclic Covering Morphisms on ¯ M ,n (2011). arXiv:1105.0655 [math.AG].[Fed13] , New nef divisors on M ,n (2013). arXiv:1308.5993 [math.AG].[Gia13] Noah Giansiracusa, Conformal blocks and rational normal curves , Journal of Algebraic Geometry, (2013), 773–793.[GG12] Noah Giansiracusa and Angela Gibney, The cone of type A, level 1 conformal block divisors , Adv.Math. (2012), 798–814.[GJMS13] Angela Gibney, David Jensen, Han-Bom Moon, and David Swinarski, Veronese quotient modelsof M ,n and conformal blocks , Michigan Math Journal (2013), 721–751.[Swi10] David Swinarski, ConformalBlocks : a Macaulay2 package for computing conformal block divisors sl conformal block divisors and the nef cone of ¯ M ,n , Exp. Math, to appear, (2011).arXiv:1107.5331 [math.AG].[TUY89] Akihiro Tsuchiya, Kenji Ueno, and Yasuhiko Yamada, Conformal field theory on universal familyof stable curves with gauge symmetries , Integrable systems in quantum field theory and statisticalmechanics, 1989, pp. 459–566. Department of Mathematics, University of Georgia, Athens, GA, 30602 E-mail address ::