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Equations of $\,\overline{{M}}_{0,n}$ | Tomesphere

arXiv:1703.07439·math.AG·March 23, 2017

Equations of $\,\overline{{M}}_{0,n}$

Leonid Monin, Julie Rana

TL;DR

This paper provides explicit polynomial equations in the Cox ring that define the moduli space of stable n-pointed rational curves, , and proves their correctness for small n, advancing understanding of its algebraic structure.

Contribution

It introduces explicit equations in the Cox ring for , with proofs for n=5 to 8 and a cohomological embedding for n.

Findings

01

Equations generate the ideal for n=5,6,7,8.

02

is embedded in projective space via the log canonical system.

03

Cohomological proof confirms the embedding for n.

Abstract

Following work of Keel and Tevelev, we give explicit polynomials in the Cox ring of $P^{1} \times \dots \times P^{n - 3}$ that, conjecturally, determine $\overline{M}_{0, n}$ as a subscheme. Using Macaulay2, we prove that these equations generate the ideal for $n = 5, 6, 7, 8$ . For $n \leq 6$ we give a cohomological proof that these polynomials realize $\overline{M}_{0, n}$ as a projective variety, embedded in $P^{(n - 2)! - 1}$ by the complete log canonical linear system.

Figures1

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Equations152

[w_{0}^{(i)} (w_{0}^{(j)} - w_{i + 1}^{(j)}) w_{0}^{(j)} w_{1}^{(i)} (w_{1}^{(j)} - w_{i + 1}^{(j)}) w_{1}^{(j)} \dots \dots w_{i}^{(i)} (w_{i}^{(j)} - w_{i + 1}^{(j)}) w_{i}^{(j)}]

[w_{0}^{(i)} (w_{0}^{(j)} - w_{i + 1}^{(j)}) w_{0}^{(j)} w_{1}^{(i)} (w_{1}^{(j)} - w_{i + 1}^{(j)}) w_{1}^{(j)} \dots \dots w_{i}^{(i)} (w_{i}^{(j)} - w_{i + 1}^{(j)}) w_{i}^{(j)}]

t_{0} t_{1} - t_{0} t_{4} + t_{2} t_{3} - t_{1} t_{2}, t_{0} t_{4} - t_{3} t_{4} + t_{3} t_{5} - t_{1} t_{5}, t_{1} t_{3} - t_{0} t_{4}, t_{2} t_{3} - t_{0} t_{5}, t_{2} t_{4} - t_{1} t_{5} .

t_{0} t_{1} - t_{0} t_{4} + t_{2} t_{3} - t_{1} t_{2}, t_{0} t_{4} - t_{3} t_{4} + t_{3} t_{5} - t_{1} t_{5}, t_{1} t_{3} - t_{0} t_{4}, t_{2} t_{3} - t_{0} t_{5}, t_{2} t_{4} - t_{1} t_{5} .

(q_{1}, \dots, q_{n}) = (g (p_{1}), \dots, g (p_{n})) .

(q_{1}, \dots, q_{n}) = (g (p_{1}), \dots, g (p_{n})) .

codim (δ (Γ)) = # V (Γ) - 1.

codim (δ (Γ)) = # V (Γ) - 1.

{\frac{d x}{( x - p _{1} ) ( x - p _{2} )}, \dots, \frac{d x}{( x - p _{1} ) ( x - p _{n - 1} )}} .

{\frac{d x}{( x - p _{1} ) ( x - p _{2} )}, \dots, \frac{d x}{( x - p _{1} ) ( x - p _{n - 1} )}} .

ψ_{n} ∣_{F} : F ↪ P^{n - 3}, x \mapsto [\frac{p _{1} - p _{2}}{x - p _{2}} : \dots : \frac{p _{1} - p _{n - 1}}{x - p _{n - 1}}] .

ψ_{n} ∣_{F} : F ↪ P^{n - 3}, x \mapsto [\frac{p _{1} - p _{2}}{x - p _{2}} : \dots : \frac{p _{1} - p _{n - 1}}{x - p _{n - 1}}] .

x \mapsto [\frac{p _{1} - p _{2}}{( x - p _{1} ) ( x - p _{2} )} : \dots : \frac{p _{1} - p _{n - 1}}{( x - p _{1} ) ( x - p _{n - 1} )}] .

x \mapsto [\frac{p _{1} - p _{2}}{( x - p _{1} ) ( x - p _{2} )} : \dots : \frac{p _{1} - p _{n - 1}}{( x - p _{1} ) ( x - p _{n - 1} )}] .

[C, p_{1}, \dots, p_{n}] \mapsto [\frac{p _{1} - p _{2}}{p _{i + 3} - p _{2}} : \dots : \frac{p _{1} - p _{i + 2}}{p _{i + 3} - p _{i + 2}}] .

[C, p_{1}, \dots, p_{n}] \mapsto [\frac{p _{1} - p _{2}}{p _{i + 3} - p _{2}} : \dots : \frac{p _{1} - p _{i + 2}}{p _{i + 3} - p _{i + 2}}] .

ϕ : [P^{1}, p_{1}, \dots, p_{4}] \mapsto [\frac{p _{1} - p _{2}}{p _{4} - p _{2}} : \frac{p _{1} - p _{3}}{p _{4} - p _{3}}] .

ϕ : [P^{1}, p_{1}, \dots, p_{4}] \mapsto [\frac{p _{1} - p _{2}}{p _{4} - p _{2}} : \frac{p _{1} - p _{3}}{p _{4} - p _{3}}] .

ϕ : \overline{M}_{0, 5} \to P^{1} \times P^{2} ≃ \overline{M}_{0, 4} \times P^{2}

ϕ : \overline{M}_{0, 5} \to P^{1} \times P^{2} ≃ \overline{M}_{0, 4} \times P^{2}

[P^{1}, p_{1}, \dots, p_{5}] \mapsto ([\frac{p _{1} - p _{2}}{p _{4} - p _{2}} : \frac{p _{1} - p _{3}}{p _{4} - p _{3}}], [\frac{p _{1} - p _{2}}{p _{5} - p _{2}} : \frac{p _{1} - p _{3}}{p _{5} - p _{3}} : \frac{p _{1} - p _{4}}{p _{5} - p _{4}}]) .

[P^{1}, p_{1}, \dots, p_{5}] \mapsto ([\frac{p _{1} - p _{2}}{p _{4} - p _{2}} : \frac{p _{1} - p _{3}}{p _{4} - p _{3}}], [\frac{p _{1} - p _{2}}{p _{5} - p _{2}} : \frac{p _{1} - p _{3}}{p _{5} - p _{3}} : \frac{p _{1} - p _{4}}{p _{5} - p _{4}}]) .

a_{0} b_{1} (b_{0} - b_{2}) - a_{1} b_{0} (b_{1} - b_{2}) .

a_{0} b_{1} (b_{0} - b_{2}) - a_{1} b_{0} (b_{1} - b_{2}) .

w_{j}^{(i)} = \frac{p _{1, j + 2}}{p _{i + 3, j + 2}}, where p_{i, j} = p_{i} - p_{j} .

w_{j}^{(i)} = \frac{p _{1, j + 2}}{p _{i + 3, j + 2}}, where p_{i, j} = p_{i} - p_{j} .

[w_{0}^{(i)} (w_{0}^{(j)} - w_{i + 1}^{(j)}) w_{0}^{(j)} w_{1}^{(i)} (w_{1}^{(j)} - w_{i + 1}^{(j)}) w_{1}^{(j)} \dots \dots w_{i}^{(i)} (w_{i}^{(j)} - w_{i + 1}^{(j)}) w_{i}^{(j)}]

[w_{0}^{(i)} (w_{0}^{(j)} - w_{i + 1}^{(j)}) w_{0}^{(j)} w_{1}^{(i)} (w_{1}^{(j)} - w_{i + 1}^{(j)}) w_{1}^{(j)} \dots \dots w_{i}^{(i)} (w_{i}^{(j)} - w_{i + 1}^{(j)}) w_{i}^{(j)}]

(w_{r}^{(j)} - w_{i + 1}^{(j)}) w_{s}^{(j)} w_{r}^{(i)} = (w_{s}^{(j)} - w_{i + 1}^{(j)}) w_{r}^{(j)} w_{s}^{(i)} .

(w_{r}^{(j)} - w_{i + 1}^{(j)}) w_{s}^{(j)} w_{r}^{(i)} = (w_{s}^{(j)} - w_{i + 1}^{(j)}) w_{r}^{(j)} w_{s}^{(i)} .

w_{r}^{(j)} - w_{i + 1}^{(j)}

w_{r}^{(j)} - w_{i + 1}^{(j)}

(w_{r}^{(j)} - w_{i + 1}^{(j)}) w_{s}^{(j)} w_{r}^{(i)}

(w_{r}^{(j)} - w_{i + 1}^{(j)}) w_{s}^{(j)} w_{r}^{(i)}

i = 1 \sum n - 4 (n - 3 - i) (2 i + 1) = (4 n - 1) .

i = 1 \sum n - 4 (n - 3 - i) (2 i + 1) = (4 n - 1) .

C [a_{0}, a_{1}, b_{0}, b_{1}, b_{2}, c_{0}, c_{1}, c_{2}, c_{3}, d_{0}, d_{1}, d_{2}, d_{3}, d_{4}] .

C [a_{0}, a_{1}, b_{0}, b_{1}, b_{2}, c_{0}, c_{1}, c_{2}, c_{3}, d_{0}, d_{1}, d_{2}, d_{3}, d_{4}] .

[a_{0} (b_{0} - b_{2}) b_{0} a_{1} (b_{1} - b_{2}) b_{1}], [a_{0} (c_{0} - c_{2}) c_{0} a_{1} (c_{1} - c_{2}) c_{1}], [a_{0} (d_{0} - d_{2}) d_{0} a_{1} (d_{1} - d_{2}) d_{1}]

[a_{0} (b_{0} - b_{2}) b_{0} a_{1} (b_{1} - b_{2}) b_{1}], [a_{0} (c_{0} - c_{2}) c_{0} a_{1} (c_{1} - c_{2}) c_{1}], [a_{0} (d_{0} - d_{2}) d_{0} a_{1} (d_{1} - d_{2}) d_{1}]

[b_{0} (c_{0} - c_{3}) c_{0} b_{1} (c_{1} - c_{3}) c_{1} b_{2} (c_{2} - c_{3}) c_{2}], [b_{0} (d_{0} - d_{3}) d_{0} b_{1} (d_{1} - d_{3}) d_{1} b_{2} (d_{2} - d_{3}) d_{2}]

[b_{0} (c_{0} - c_{3}) c_{0} b_{1} (c_{1} - c_{3}) c_{1} b_{2} (c_{2} - c_{3}) c_{2}], [b_{0} (d_{0} - d_{3}) d_{0} b_{1} (d_{1} - d_{3}) d_{1} b_{2} (d_{2} - d_{3}) d_{2}]

[c_{0} (d_{0} - d_{4}) d_{0} c_{1} (d_{1} - d_{4}) d_{1} c_{2} (d_{2} - d_{4}) d_{2} c_{3} (d_{3} - d_{4}) d_{3}] .

[c_{0} (d_{0} - d_{4}) d_{0} c_{1} (d_{1} - d_{4}) d_{1} c_{2} (d_{2} - d_{4}) d_{2} c_{3} (d_{3} - d_{4}) d_{3}] .

c_{2} d_{2} d_{3} - c_{3} d_{2} d_{3} + c_{3} d_{2} d_{4} - c_{2} d_{3} d_{4}, c_{1} d_{1} d_{3} - c_{3} d_{1} d_{3} + c_{3} d_{1} d_{4} - c_{1} d_{3} d_{4} c_{0} d_{0} d_{3} - c_{3} d_{0} d_{3} + c_{3} d_{0} d_{4} - c_{0} d_{3} d_{4}, c_{1} d_{1} d_{2} - c_{2} d_{1} d_{2} + c_{2} d_{1} d_{4} - c_{1} d_{2} d_{4} b_{1} d_{1} d_{2} - b_{2} d_{1} d_{2} + b_{2} d_{1} d_{3} - b_{1} d_{2} d_{3}, c_{0} d_{0} d_{2} - c_{2} d_{0} d_{2} + c_{2} d_{0} d_{4} - c_{0} d_{2} d_{4} b_{0} d_{0} d_{2} - b_{2} d_{0} d_{2} + b_{2} d_{0} d_{3} - b_{0} d_{2} d_{3}, c_{0} d_{0} d_{1} - c_{1} d_{0} d_{1} + c_{1} d_{0} d_{4} - c_{0} d_{1} d_{4} b_{0} d_{0} d_{1} - b_{1} d_{0} d_{1} + b_{1} d_{0} d_{3} - b_{0} d_{1} d_{3}, a_{0} d_{0} d_{1} - a_{1} d_{0} d_{1} + a_{1} d_{0} d_{2} - a_{0} d_{1} d_{2} b_{1} c_{1} c_{2} - b_{2} c_{1} c_{2} + b_{2} c_{1} c_{3} - b_{1} c_{2} c_{3}, b_{0} c_{0} c_{2} - b_{2} c_{0} c_{2} + b_{2} c_{0} c_{3} - b_{0} c_{2} c_{3} b_{0} c_{0} c_{1} - b_{1} c_{0} c_{1} + b_{1} c_{0} c_{3} - b_{0} c_{1} c_{3}, a_{0} c_{0} c_{1} - a_{1} c_{0} c_{1} + a_{1} c_{0} c_{2} - a_{0} c_{1} c_{2} a_{0} b_{0} b_{1} - a_{1} b_{0} b_{1} + a_{1} b_{0} b_{2} - a_{0} b_{1} b_{2}

c_{2} d_{2} d_{3} - c_{3} d_{2} d_{3} + c_{3} d_{2} d_{4} - c_{2} d_{3} d_{4}, c_{1} d_{1} d_{3} - c_{3} d_{1} d_{3} + c_{3} d_{1} d_{4} - c_{1} d_{3} d_{4} c_{0} d_{0} d_{3} - c_{3} d_{0} d_{3} + c_{3} d_{0} d_{4} - c_{0} d_{3} d_{4}, c_{1} d_{1} d_{2} - c_{2} d_{1} d_{2} + c_{2} d_{1} d_{4} - c_{1} d_{2} d_{4} b_{1} d_{1} d_{2} - b_{2} d_{1} d_{2} + b_{2} d_{1} d_{3} - b_{1} d_{2} d_{3}, c_{0} d_{0} d_{2} - c_{2} d_{0} d_{2} + c_{2} d_{0} d_{4} - c_{0} d_{2} d_{4} b_{0} d_{0} d_{2} - b_{2} d_{0} d_{2} + b_{2} d_{0} d_{3} - b_{0} d_{2} d_{3}, c_{0} d_{0} d_{1} - c_{1} d_{0} d_{1} + c_{1} d_{0} d_{4} - c_{0} d_{1} d_{4} b_{0} d_{0} d_{1} - b_{1} d_{0} d_{1} + b_{1} d_{0} d_{3} - b_{0} d_{1} d_{3}, a_{0} d_{0} d_{1} - a_{1} d_{0} d_{1} + a_{1} d_{0} d_{2} - a_{0} d_{1} d_{2} b_{1} c_{1} c_{2} - b_{2} c_{1} c_{2} + b_{2} c_{1} c_{3} - b_{1} c_{2} c_{3}, b_{0} c_{0} c_{2} - b_{2} c_{0} c_{2} + b_{2} c_{0} c_{3} - b_{0} c_{2} c_{3} b_{0} c_{0} c_{1} - b_{1} c_{0} c_{1} + b_{1} c_{0} c_{3} - b_{0} c_{1} c_{3}, a_{0} c_{0} c_{1} - a_{1} c_{0} c_{1} + a_{1} c_{0} c_{2} - a_{0} c_{1} c_{2} a_{0} b_{0} b_{1} - a_{1} b_{0} b_{1} + a_{1} b_{0} b_{2} - a_{0} b_{1} b_{2}

w_{k + 1}^{(m)} w_{l + 1}^{(m)} (w_{i}^{(k)} w_{j}^{(l)} - w_{j}^{(k)} w_{i}^{(l)}) + w_{l + 1}^{(m)} w_{j}^{(m)} (w_{j}^{(k)} w_{i}^{(l)} - w_{i}^{(k)} w_{i}^{(l)}) + + w_{j}^{(m)} w_{k + 1}^{(m)} (w_{i}^{(k)} w_{i}^{(l)} - w_{i}^{(k)} w_{j}^{(l)}) = 0.

w_{k + 1}^{(m)} w_{l + 1}^{(m)} (w_{i}^{(k)} w_{j}^{(l)} - w_{j}^{(k)} w_{i}^{(l)}) + w_{l + 1}^{(m)} w_{j}^{(m)} (w_{j}^{(k)} w_{i}^{(l)} - w_{i}^{(k)} w_{i}^{(l)}) + + w_{j}^{(m)} w_{k + 1}^{(m)} (w_{i}^{(k)} w_{i}^{(l)} - w_{i}^{(k)} w_{j}^{(l)}) = 0.

w_{k - 2}^{(m - 3)} w_{l - 2}^{(m - 3)} (w_{i - 2}^{(k - 3)} w_{j - 2}^{(l - 3)} - w_{j - 2}^{(k - 3)} w_{i - 2}^{(l - 3)}) +

w_{k - 2}^{(m - 3)} w_{l - 2}^{(m - 3)} (w_{i - 2}^{(k - 3)} w_{j - 2}^{(l - 3)} - w_{j - 2}^{(k - 3)} w_{i - 2}^{(l - 3)}) +

+ w_{l - 2}^{(m - 3)} w_{j - 2}^{(m - 3)} (w_{j - 2}^{(k - 3)} w_{i - 2}^{(l - 3)} - w_{i - 2}^{(k - 3)} w_{i - 2}^{(l - 3)}) +

+ w_{l - 2}^{(m - 3)} w_{j - 2}^{(m - 3)} (w_{j - 2}^{(k - 3)} w_{i - 2}^{(l - 3)} - w_{i - 2}^{(k - 3)} w_{i - 2}^{(l - 3)}) +

+ w_{j - 2}^{(m - 3)} w_{k - 2}^{(m - 3)} (w_{i - 2}^{(k - 3)} w_{i - 2}^{(l - 3)} - w_{i - 2}^{(k - 3)} w_{j - 2}^{(l - 3)}) = 0.

+ w_{j - 2}^{(m - 3)} w_{k - 2}^{(m - 3)} (w_{i - 2}^{(k - 3)} w_{i - 2}^{(l - 3)} - w_{i - 2}^{(k - 3)} w_{j - 2}^{(l - 3)}) = 0.

w_{k - 2}^{(m - 3)} w_{l - 2}^{(m - 3)} (w_{i - 2}^{(k - 3)} w_{j - 2}^{(l - 3)} - w_{j - 2}^{(k - 3)} w_{i - 2}^{(l - 3)}) = A \frac{p _{k, l}}{p _{m, l} p _{m, k} p _{k, j} p _{l, j}};

w_{k - 2}^{(m - 3)} w_{l - 2}^{(m - 3)} (w_{i - 2}^{(k - 3)} w_{j - 2}^{(l - 3)} - w_{j - 2}^{(k - 3)} w_{i - 2}^{(l - 3)}) = A \frac{p _{k, l}}{p _{m, l} p _{m, k} p _{k, j} p _{l, j}};

w_{l - 2}^{(m - 3)} w_{j - 2}^{(m - 3)} (w_{j - 2}^{(k - 3)} w_{i - 2}^{(l - 3)} - w_{i - 2}^{(k - 3)} w_{i - 2}^{(l - 3)}) = A \frac{1}{p _{m, l} p _{m, j} p _{k, j}};

w_{l - 2}^{(m - 3)} w_{j - 2}^{(m - 3)} (w_{j - 2}^{(k - 3)} w_{i - 2}^{(l - 3)} - w_{i - 2}^{(k - 3)} w_{i - 2}^{(l - 3)}) = A \frac{1}{p _{m, l} p _{m, j} p _{k, j}};

w_{j - 2}^{(m - 3)} w_{k - 2}^{(m - 3)} (w_{i - 2}^{(k - 3)} w_{i - 2}^{(l - 3)} - w_{i - 2}^{(k - 3)} w_{j - 2}^{(l - 3)}) = - A \frac{1}{p _{m, j} p _{m, k} p _{l, j}};

w_{j - 2}^{(m - 3)} w_{k - 2}^{(m - 3)} (w_{i - 2}^{(k - 3)} w_{i - 2}^{(l - 3)} - w_{i - 2}^{(k - 3)} w_{j - 2}^{(l - 3)}) = - A \frac{1}{p _{m, j} p _{m, k} p _{l, j}};

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TopicsDifferential Equations and Boundary Problems · advanced mathematical theories · Differential Equations and Numerical Methods

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11institutetext: Leonid Monin 22institutetext: University of Toronto, 40 St. George Street, Toronto, Ontario, 22email: [email protected] 33institutetext: Julie Rana 44institutetext: University of Minnesota, 206 Church St. SE, Minneapolis MN, 44email: [email protected]

Equations of $\,\operatorname{\overline{{M}}}_{0,n}$

Leonid Monin

Julie Rana

Abstract

Following work of Keel and Tevelev, we give explicit polynomials in the Cox ring of $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ that, conjecturally, determine $\operatorname{\overline{{M}}}_{0,n}$ as a subscheme. Using Macaulay2, we prove that these equations generate the ideal for $n=5,6,7,8$ . For $n\leq 6$ we give a cohomological proof that these polynomials realize $\operatorname{\overline{{M}}}_{0,n}$ as a projective variety, embedded in $\mathbb{P}^{(n-2)!-1}$ by the complete log canonical linear system.

1 Introduction

Let $\operatorname{{M}}_{0,n}$ be the moduli space of genus zero curves with $n$ marked points, and $\operatorname{\overline{{M}}}_{0,n}$ its Deligne-Mumford-Knudsen compactification DM (69). These spaces were first studied in detail by Knudsen and Mumford KM (76); Knu83a ; Knu83b .

In Kap93b , Kapranov constructed $\operatorname{\overline{{M}}}_{0,n}$ as a Chow quotient of the Grassmannian $G(2,n)$ , which allowed him to present $\operatorname{\overline{{M}}}_{0,n}$ as a sequence of blowups of $\mathbb{P}^{n-3}$ . Following this, Keel and Tevelev in KT (09) described an embedding of $\operatorname{\overline{{M}}}_{0,n}$ into the space of sections of particular characteristic classes on $\operatorname{\overline{{M}}}_{0,n}$ , as well as existence of an embedding $\phi:\operatorname{\overline{{M}}}_{0,n}\to\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ . We obtain a list of equations satisfied by $\phi(\operatorname{\overline{{M}}}_{0,n})$ in the Cox ring of $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ .

Lemma 1

Consider the embedding $\phi:\operatorname{\overline{{M}}}_{0,n}\to\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ . Let $w_{0}^{(i)},\ldots,w_{i}^{(i)}$ be homogeneous coordinates on the $i^{\textrm{th}}$ component. With a choice of coordinates, the image of $\phi$ satisfies the $\binom{n-1}{4}$ equations given by the $2\times 2$ minors of the matrices

[TABLE]

for all $1\leq i<j\leq n-3$ .

Conjecture 1

Let $I_{n}$ be the prime ideal in the Cox ring of $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ that defines the embedding $\phi(\operatorname{\overline{{M}}}_{0,n})$ scheme-theoretically. Let $J_{n}$ be the ideal in the Cox ring of $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ generated by the equations of Lemma 1. Then $I_{n}$ is the unique $B$ -saturation of $J_{n}$ , where $B=\cap_{i=1}^{n-3}\langle w_{0}^{(i)},\ldots,w_{i}^{(i)}\rangle$ . The ideal $I_{n}$ is minimally generated by $\binom{n-1}{d+1}$ polynomials of degree $d$ , for $d=3,4,5,\ldots,n-2.$ The degree of $I_{n}$ is $(2n-7)!!$ , the number of trivalent phylogenetic trees on $n-1$ leaves. The lexicographic initial monomial ideals are square-free and Cohen-Macaulay.

We verified Conjecture 1 for $n=5,6,7,8$ using Macaulay2. In particular, we partially answered the question posed in (Sturm, , Problem 8 on Curves).

In Section 4, we provide a list of degree $4$ polynomials contained in $I_{n}$ and describe a conjectural method to count the number of minimal generators of $I_{n}$ in any degree. The combinatorial description of the ideal of relations of the invariant ring of $\operatorname{\overline{{M}}}_{0,n}$ presented in HMSV (09) offers another possibly promising method.

In a slightly different direction, we consider the embedding $\Phi$ of $\operatorname{\overline{{M}}}_{0,n}$ into $\mathbb{P}^{(n-3)!-1}$ given by the $\kappa$ class. By work of Keel and Tevelev KT (09), the ideal defining $\operatorname{\overline{{M}}}_{0,n}$ as a subscheme of $\mathbb{P}^{(n-3)!-1}$ is generated by the polynomials defining the Segre embedding $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}\to\mathbb{P}^{(n-3)!-1}$ , together with the preimages in the Cox ring of $\mathbb{P}^{(n-3)!-1}$ of the polynomials in $I_{n}$ of degree $(2,2,\ldots,2)$ . For example, in the case of $\operatorname{\overline{{M}}}_{0,5}$ , we have

Corollary 1

The ideal of the embedding of $\operatorname{\overline{{M}}}_{0,5}$ into $\mathbb{P}^{5}$ via the $\kappa$ class is generated by the five quadrics

[TABLE]

Using the Chow quotient description of $\operatorname{\overline{{M}}}_{0,n}$ due to Kapranov Kap93a , Gibney and Maclagan GM (11) provide equations for $\operatorname{\overline{{M}}}_{0,5}\subset\mathbb{P}^{21}$ . Many of the listed equations in GM (11) are linear, so they have effectively given an embedding of $\operatorname{\overline{{M}}}_{0,5}$ into $\mathbb{P}^{5}$ . Using Macaulay2, we were able to eliminate variables in such a way that the resulting embedding is a nonsingular variety of dimension 2 given by five quadrics in $\mathbb{P}^{5}$ . Beyond this, it is not yet clear how these equations relate to ours.

The result of Keel and Tevelev motivates the following

Conjecture 2

The ideal $J_{n}$ contains all polynomials of degree $(2,2,\ldots,2)$ in $I_{n}$ , and is therefore enough to determine the equations of $\Phi(\operatorname{\overline{{M}}}_{0,n})$ in the Cox ring of $\mathbb{P}^{(n-3)!-1}$ ideal-theoretically.

In Section 5, we prove Conjecture 2 for $\operatorname{\overline{{M}}}_{0,5}$ and $\operatorname{\overline{{M}}}_{0,6}$ using cohomological techniques developed in KT (09). We hope that these techniques can be extended to larger values of $n$ . In particular, we have

Theorem 1.1

Let $J_{6}$ be the ideal generated by the five polynomials of Lemma 4. Then the ideal $\tilde{I}_{6}$ generated by polynomials of degree $(d,d,d)$ in $I_{6}$ is generated by the degree $(2,2,2)$ polynomials in $J_{6}$ . Equivalently, the embedding of $\operatorname{\overline{{M}}}_{0,n}$ in $\mathbb{P}^{23}$ defined by the $\kappa$ class is generated by the polynomials of degree $(2,2,2)$ in $J_{6}$ and the Segre relations.

The paper is structured as follows. In Section 2, we give some background about the moduli spaces $\operatorname{\overline{{M}}}_{0,n}$ . In Section 3, we describe in detail the embedding $\phi:\operatorname{\overline{{M}}}_{0,n}\to\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ which we will use in later sections to find equations of $\operatorname{\overline{{M}}}_{0,n}$ . In particular, away from the boundary of $\operatorname{\overline{{M}}}_{0,n}$ , we give a parametrization of $\phi$ that extends to the full moduli space, and provide a geometric interpretation of $\phi$ in the cases $n=4$ and $n=5$ . In Section 4, we give explicit equations satisfied by $\phi(\operatorname{\overline{{M}}}_{0,n})$ that are forced by the parametrization. In Sections 5 and 6 we recall the cohomological machinery developed in KT (09) and use it to prove that $J_{5}$ and $J_{6}$ contain all polynomials of degree $(2,2,\ldots,2)$ in the respective Cox rings.

2 Background on the moduli space $\operatorname{\overline{{M}}}_{0,n}$

We begin with a brief introduction to the moduli space of pointed rational curves. For more details, we recommend the lectures and lecture notes of Cavalieri Cav (16). For those new to the theory of stable curves, we recommend HM (98).

For $n\geq 3$ , the moduli space $\operatorname{{M}}_{0,n}$ parametrizes ordered $n$ -tuples of distinct points on $\mathbb{P}^{1}$ . Two $n$ -tuples $(p_{1},\ldots,p_{n})$ and $(q_{1},\ldots,q_{n})$ are equivalent if there exists a projective transformation $g\in\operatorname{PGL}(2,\mathbb{C})$ such that

[TABLE]

Since a projective transformation can map three points in $\mathbb{P}^{1}$ to any other three points and is uniquely determined by their image, the dimension of $\operatorname{{M}}_{0,n}$ equals $n-3$ .

The space $\operatorname{{M}}_{0,n}$ is not compact because the points $p_{i}$ are distinct. There are a number of compactifications of $\operatorname{{M}}_{0,n}$ , including those described by Losev-Manin LM (00) and Keel Kee (92). But the first and most well-known is $\operatorname{\overline{{M}}}_{0,n}$ , the Deligne-Mumford-Knudsen compactification, described explicitly by Kapranov Kap93a ; Kap93b . The moduli space $\operatorname{\overline{{M}}}_{0,n}$ parametrizes stable $n$ -pointed rational curves.

Definition 1

A stable $n$ -pointed rational curve is a tuple $(C,p_{1},\ldots,p_{n})$ , where

$C$ is a connected curve of arithmetic genus [math] with at most simple nodal singularities; 2. 2.

$p_{1},\ldots,p_{n}$ are distinct nonsingular points on $C$ ; 3. 3.

each irreducible component of $C$ has at least three special points (either marked points or nodes).

The dual graph of a stable curve $(C,p_{1},\ldots,p_{n})$ is defined as a graph having a vertex for each irreducible component, an edge between two vertices for each point of intersection of the corresponding components, and for each marked point, a labeled half edge attached to the appropriate vertex. Since $C$ has arithmetic genus [math], the dual graph is a tree.

The boundary $\operatorname{\overline{{M}}}_{0,n}\setminus\operatorname{{M}}_{0,n}$ is a normal crossing divisor with a natural stratification by the dual graphs. The codimension of the stratum $\delta(\Gamma)$ in $\operatorname{\overline{{M}}}_{0,n}$ corresponding to the dual graph $\Gamma$ is one less than the number of vertices of $\Gamma$ :

[TABLE]

Thus, each divisorial component of the boundary of $\operatorname{\overline{{M}}}_{0,n}$ corresponds to stable curves with dual graph $\Gamma$ corresponding to a partition of $\{1,\ldots,n\}$ into two disjoint sets $I$ and $I^{c}$ , each of cardinality at least 2. Given such a partitition, we denote the corresponding irreducible boundary divisor of $\operatorname{\overline{{M}}}_{0,n}$ by $\delta_{I}$ . Note that two divisors $\delta_{I}$ and $\delta_{J}$ intersect in $\operatorname{\overline{{M}}}_{0,n}$ if and only if $I\subset J$ , $I\subset J^{c}$ , $J\subset I$ , or $J\subset I^{c}$ .

Example 1

Consider the graph whose vertices correspond to the irreducible boundary divisors in $\operatorname{\overline{{M}}}_{0,5}$ . Two vertices are joined by an edge if the corresponding boundary divisors intersect (see Figure 1). Originally a purely combinatorial construction, the Petersen graph shown in Figure 1 is in fact the link of the tropical moduli space $M^{\textrm{trop}}_{0,5}$ . For a brief introduction to the moduli space $M^{\textrm{trop}}_{g,n}$ , we recommend the lectures and lecture notes of Melody Chan Cha . For a thorough introduction to tropical geometry, see MS (15).

Remark 1

A general philosophy is that birational models of a given compactified moduli space should provide alternate compactifications which themselves have modular interpretations. From this perspective, describing the birational geometry of $\operatorname{\overline{{M}}}_{0,n}$ is not only interesting in its own right, but also provides a window into current research in moduli theory. As a first step in this direction, in HM (82), Harris and Mumford proved that the moduli spaces $\operatorname{\overline{{M}}}_{g,n}$ are of general type for large enough $g$ . Thus, understanding the birational geometry of $\operatorname{\overline{{M}}}_{g,n}$ boils down to describing all ample divisors on the moduli space. A long-standing conjecture of Fulton and Faber, the so-called “F-conjecture”, describes the ample cone of $\operatorname{\overline{{M}}}_{g,n}$ ; see for example GKM (02) in which the F-conjecture is reduced to the genus [math] case. Notably, in HK (00), Hu and Keel conjectured that $\operatorname{\overline{{M}}}_{0,n}$ is a Mori Dream Space, a result that would have implied the F-conjecture. This was recently disproved for $n>133$ by Castravet and Tevelev CT (15); their techniques were quickly extended to $n>13$ in GK . We recommend the excellent survey FS (13) for those interested in learning more about birational models and alternative compactifications of $\operatorname{\overline{{M}}}_{g,n}$ .

3 The embedding $\operatorname{\overline{{M}}}_{0,n}$ in $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$

We begin with a description of an embedding $\phi:\operatorname{\overline{{M}}}_{0,n}\hookrightarrow\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ , followed by a parametrization of $\phi$ on the interior of $\operatorname{\overline{{M}}}_{0,n}$ which extends to the full moduli space. We then give a geometric interpretation of $\phi$ in the cases $\operatorname{\overline{{M}}}_{0,4}$ and $\operatorname{\overline{{M}}}_{0,5}$ . This realizes $\operatorname{\overline{{M}}}_{0,5}$ in Theorem 2 as a pencil of conics in $\mathbb{P}^{1}\times\mathbb{P}^{2}$ . Finally, we use the parametrization to list a set of equations satisfied by $\phi(\operatorname{\overline{{M}}}_{0,n})\subset\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ .

We recall two well-studied maps from $\operatorname{\overline{{M}}}_{0,n}$ . The first, the forgetful map $\pi_{n}:\operatorname{\overline{{M}}}_{0,n}\to\operatorname{\overline{{M}}}_{0,n-1}$ , is given by forgetting the last point of $[C,p_{1},\ldots p_{n}]$ and stabilizing the curve. That is, $\pi_{n}$ contracts the components of $C$ that have less than three special points among $p_{1},\ldots,p_{n-1}$ , and remembers the points of intersection.

For the second, let $\mathbb{L}_{i}$ be the line bundle on $\operatorname{\overline{{M}}}_{0,n}$ whose fiber over a point $[C,p_{1},\ldots,p_{n}]$ is the cotangent space of $\mathbb{P}^{1}$ at $p_{i}$ . Define $\psi_{i}=c_{1}(\mathbb{L}_{i})$ to be the first Chern class of $\mathbb{L}_{i}$ . The Kapranov map $\psi_{n}$ is the rational map given by the linear system $|\psi_{n}|$ . This map was first described in detail by Kapranov Kap93b , who proved in particular that $\psi_{n}:\operatorname{\overline{{M}}}_{0,n}\to\mathbb{P}^{n-3}$ .

Theorem 3.1

[KT, Cor 2.7] The map $\Phi=(\pi_{n},\psi_{n}):\operatorname{\overline{{M}}}_{0,n}\rightarrow\operatorname{\overline{{M}}}_{0,n-1}\times\mathbb{P}^{n-3}$ is a closed embedding.

Corollary 2

We have a closed embedding $\phi:\operatorname{\overline{{M}}}_{0,n}\hookrightarrow\mathbb{P}^{1}\times\mathbb{P}^{2}\times\ldots\times\mathbb{P}^{n-3}$ .

Proof

Apply Theorem 3.1 successively. ∎

Let us describe the map $\phi$ of Corollary 2 explicitly. Since $\phi$ is a closed embedding, it is enough to describe it only on the smooth part $\operatorname{{M}}_{0,n}$ of $\operatorname{\overline{{M}}}_{0,n}$ , which is an open, dense subset of $\operatorname{\overline{{M}}}_{0,n}$ . To this end, consider the restriction of $\pi_{n}$ to $\operatorname{{M}}_{0,n}$ and let $F=\pi_{n}^{-1}([\mathbb{P}^{1},p_{1},\ldots,p_{n-1}])\simeq\mathbb{P}^{1}\backslash\{p_{1},\ldots,p_{n-1}\}$ be the fiber over a point in $\operatorname{{M}}_{0,n-1}$ . We have the following description of the line bundle $\mathbb{L}_{n}$ over the fiber $F$ .

Lemma 2

KT (09)** We have $\mathbb{L}_{n}|_{F}=\omega_{\mathbb{P}^{1}}(p_{1}+\cdots+p_{n-1})$ . In particular, $\psi_{n}|_{F}=K_{\mathbb{P}^{1}}+p_{1}+\cdots+p_{n-1}.$

Following Lemma 2, we obtain a basis of $H^{0}(F,K_{F}+p_{1}+\cdots+p_{n-1})$ .

Lemma 3

The vector space $H^{0}(F,K_{F}+p_{1}+\cdots+p_{n-1})$ has dimension $n-2$ . A basis is given by the one-forms

[TABLE]

Proof

Since the canonical class of $F\simeq\mathbb{P}^{1}$ is $K_{F}=-2[\textrm{pt}]$ , the dimension of the space of global sections follows from, for example, Riemann-Roch. Since the $n-2$ one-forms listed have two poles, each at different pairs of points, they are linearly independent, and so form a basis. ∎

We thus obtain an explicit parametrization of the map $\psi_{n}|_{F}:F\hookrightarrow\mathbb{P}^{n-3}$ .

Proposition 1

Let $F\simeq\mathbb{P}^{1}\backslash\{p_{1},\ldots,p_{n-1}\}$ be the fiber of the map $\pi_{n}$ over the point $[\mathbb{P}^{1},p_{1},\ldots,p_{n-1}]\in\operatorname{{M}}_{0,n-1}$ . The restriction of $\psi_{n}$ to $F$ is given in coordinates by

[TABLE]

Proof

Using a properly rescaled basis from Lemma 3, we can write the map as

[TABLE]

Multiplying through by $x-p_{1}$ gives the result. ∎

By Corollary 2, we can extend $\psi_{n}$ uniquely to all of $\operatorname{\overline{{M}}}_{0,n}$ . By our choice of basis, the points $p_{1},\ldots,p_{n-1}\in\bar{F}$ map to the coordinate points of $\mathbb{P}^{n-3}$ , in particular to points in general position. Thus, $\psi_{n}(\bar{F})$ is a degree $n-3$ curve in $\mathbb{P}^{n-3}$ , i.e. a (generically) smooth rational normal curve passing through these $n-1$ fixed points.

Taking the parameter $x$ to be $p_{n}$ , the map $\phi:\operatorname{\overline{{M}}}_{0,n}\rightarrow\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ is given in the $i^{\textrm{th}}$ component by

[TABLE]

Example 2

We describe the embedding $\phi:\operatorname{\overline{{M}}}_{0,4}\to\mathbb{P}^{1}$ explicitly. Away from $p_{2},p_{3},p_{4}$ , we have

[TABLE]

Thus, away from the boundary of $\operatorname{\overline{{M}}}_{0,4}$ , we see that $\phi$ is an isomorphism mapping a 4-tuple of distinct points on $\mathbb{P}^{1}$ to their cross-ratio.

The boundary of $\operatorname{\overline{{M}}}_{0,4}$ consists of three points: $\delta_{1,2}$ , $\delta_{1,3}$ , and $\delta_{1,4}$ (see Fig. 2). By taking limits $p_{1}\to p_{2}$ , $p_{1}\to p_{3}$ , and $p_{1}\to p_{4}$ , respectively, we see that these boundary points map under $\phi$ to the points $[0:1],[1:0],$ and $[1:1]$ , respectively.

Example 3

For $n=5$ the embedding above has the form

[TABLE]

[TABLE]

The forgetful map $\pi_{5}$ restricts to $\operatorname{\overline{{M}}}_{0,5}$ the projection of $\operatorname{\overline{{M}}}_{0,4}\times\mathbb{P}^{2}$ onto the first factor, so the fiber of $\pi_{5}$ over any point in $\operatorname{{M}}_{0,4}\simeq\mathbb{P}^{1}\setminus\{[0:1],[1:0],[1:1]\}$ is a smooth conic passing through four fixed points $[1:0:0]$ , $[0:1:0]$ , $[0:0:1]$ , and $[1:1:1]$ in a copy of $\mathbb{P}^{2}$ . The fibers over $[0:1],[1:0],[1:1]$ are the singular conics passing through these four fixed points. In particular, six boundary divisors of $\operatorname{\overline{{M}}}_{0,5}$ map to the components of three singular conics, and the remaining four map to the fibers $\mathbb{P}^{1}\times\{[1:0:0]\}$ , $\mathbb{P}^{1}\times\{[0:1:0]\}$ , $\mathbb{P}^{1}\times\{[0:0:1]\}$ , and $\mathbb{P}^{1}\times\{[1:1:1]\}$ (see Fig. 3).

Since $\operatorname{\overline{{M}}}_{0,5}$ is the pencil of conics in $\mathbb{P}^{2}$ passing through these given four points, we easily write down an embedding of $\operatorname{\overline{{M}}}_{0,5}$ in $\mathbb{P}^{1}_{[a_{0}:a_{1}]}\times\mathbb{P}^{2}_{[b_{0}:b_{1}:b_{2}]}$ . Its equation is

[TABLE]

We observe that $\psi:\operatorname{\overline{{M}}}_{0,5}\to\mathbb{P}^{2}$ is a blowdown. Indeed, since there is unique conic passing through any five points in $\mathbb{P}^{2}$ , the map $\psi:\operatorname{\overline{{M}}}_{0,5}\to\mathbb{P}^{2}$ is 1-to-1 away from $[1:0:0],[0:1:0],[0:0:1]$ , and $[1:1:1]$ . The fibers over these points are all isomorphic to $\mathbb{P}^{1}$ .

Proposition 2

With the choice of coordinates above, the ideal which defines the moduli space $\operatorname{\overline{{M}}}_{0,5}$ as a projective variety in $\mathbb{P}^{1}_{[a_{0}:a_{1}]}\times\mathbb{P}^{2}_{[b_{0}:b_{1}:b_{2}]}$ is generated by (2).

Proof

This is a direct computation consisting of checking that the variety described by this principal ideal in the multigraded ring $\mathbb{C}[a_{0},a_{1},b_{0},b_{1},b_{2}]$ is two-dimensional, smooth, and irreducible. We verified this using Macaulay2. ∎

4 Equations of $\operatorname{\overline{{M}}}_{0,n}$

We provide a list of equations contained in the ideal defining the scheme $\phi(\operatorname{\overline{{M}}}_{0,n})$ in the Cox ring of $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ . Let $w_{0}^{(i)},\ldots,w_{i}^{(i)}$ be homogeneous coordinates on the $i^{\textrm{th}}$ factor of $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ . By Proposition 1, the embedding $\phi:\operatorname{\overline{{M}}}_{0,n}\rightarrow\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ is given in coordinates by

[TABLE]

Lemma 4

The image of $\phi$ in $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ satisfies the $\binom{n-1}{4}$ polynomials given by the $2\times 2$ minors of the matrices

[TABLE]

for all $1\leq i<j\leq n-3$ . Let $J_{n}$ be the ideal generated by these polynomials. No proper subset of these polynomials forms a basis of $J_{n}$ .

Proof

The proof is direct calculation. Choose columns $r$ and $s$ . We show that

[TABLE]

Indeed, the following equalities hold:

[TABLE]

So we have

[TABLE]

A similar calculation gives that $\left(w_{s}^{(j)}-w_{i+1}^{(j)}\right)w_{r}^{(j)}w_{s}^{(i)}$ equals the same expression.

For each $i\in\{1,\ldots,n-4\}$ there are $n-3-i$ matrices with $i+1$ columns each. Thus, the number of $2\times 2$ minors is given by

[TABLE]

The polynomials have the same total degree and different initial terms under lex order, so are linearly independent over $\mathbb{C}$ . This proves the final statement. ∎

Example 4

We list the polynomials of Lemma 4 satisfied by $\operatorname{\overline{{M}}}_{0,7}$ as a subscheme of $\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{P}^{3}\times\mathbb{P}^{4}$ . The Cox ring of the product of projective spaces is

[TABLE]

The six matrices of Lemma 4 are

[TABLE]

[TABLE]

[TABLE]

Taking all $2\times 2$ minors gives the following $15$ polynomials in the ideal defining $\operatorname{\overline{{M}}}_{0,7}$ as a subscheme of $\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{P}^{3}\times\mathbb{P}^{4}$ :

[TABLE]

Conjecture 3

Let $I_{n}$ be the $B$ -saturated ideal in $\mathbb{C}[\{w_{j}^{(i)}\}]$ that defines the subscheme $\phi(\operatorname{\overline{{M}}}_{0,n})$ of $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ , where $B=\cap_{i=1}^{n-3}\langle w_{0}^{(i)},\ldots,w_{i}^{(i)}\rangle$ . Let $G$ be a minimal Gröbner basis of $I_{n}$ . The number of polynomials of degree $d$ in $G$ is $\binom{n-1}{d+1}$ .

Computations in Macaulay2 support Conjecture 3 for $4\leq n\leq 8$ . The following lemma gives additional evidence.

Lemma 5

For each choice of $0\leq i<j\leq k<l<m\leq n-3$ , the embedding of $\operatorname{\overline{{M}}}_{0,n}$ in $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ satisfies the following $\binom{n-1}{5}$ linearly independent equations of degree 4 in the Cox ring:

[TABLE]

Proof

Note that the number of equations with $j=k$ is $\binom{n-2}{5}$ and with $j\neq k$ is $\binom{n-2}{4}$ . Thus there are $\binom{n-2}{5}+\binom{n-2}{4}=\binom{n-1}{5}$ equations. Since all of the equations have different initial terms with respect to lex order, they are linearly independent.

The rest of the proof is direct calculation. For simplicity, we first shift the indices:

[TABLE]

[TABLE]

[TABLE]

Then one checks that the following identities hold:

[TABLE]

[TABLE]

[TABLE]

where $A=\frac{p_{1,i}p_{1,j}p_{1,l}p_{1,k}p_{j,i}}{p_{l,i}p_{k,i}}$ . Furthermore the following is true:

[TABLE]

which completes the proof. ∎

The equations in Lemma 5 are equivalent to the following, modulo the ideal $J_{n}$ :

[TABLE]

Indeed, we have

[TABLE]

We expect that the polynomials (4) are the minimal generators of $I_{n}$ in degree $4$ .

Example 5

According to Conjecture 3, we expect $\binom{7-1}{4+1}=6$ polynomials of degree $4$ in a reduced Gröbner basis of $I_{7}$ . Using Macaulay2, we compute the unique $B$ -saturation of $J_{7}$ . This gives six polynomials of degree $4$ in

[TABLE]

in support of Conjecture 3. These are

[TABLE]

These six polynomials correspond to the equations (4).

We note two interesting features. First, the last equation coincides with the unique equation of degree $4$ in a Gröbner basis of $I_{6}$ , and the structure of each equation is similar to this one. In general, we expect that equations of degree $d$ in a Gröbner basis for $I_{n}$ have structure similar to the unique equation of top degree in a Gröbner basis for $I_{d+2}$ . Secondly, for each pair $0\leq i<j\leq 2$ there is a unique polynomial of degree $(0,1,1,2)$ containing $b_{i}$ and $b_{j}$ .

Let $G$ be a reduced Gröbner basis of $I_{n}$ under lex order, and $G_{d}$ the subset of $G$ consisting of polynomials of degree $d$ . Fix a $\mathbb{P}^{i}$ and choose two $w_{j}^{(i)},w_{k}^{(i)}$ . For each choice of $d-2$ of the remaining spaces $\mathbb{P}^{i+1},\ldots,\mathbb{P}^{n-3}$ , we conjecture that there is a unique polynomial in $G_{d}$ of degree one in the chosen variables, other than the last occurring projective space, in which the polynomial has degree two.

If the polynomials of degree $d$ can be counted in this way, then Conjecture 3 will be true, upon application of the following combinatorial fact:

Lemma 6

[TABLE]

Proof

We rewrite the left hand side as a hypergeometric series and apply the Chu-Vandermonde Identity, see e.g. Roy (87). Let ${}_{2}F_{1}\begin{pmatrix}a&b\\ &c\end{pmatrix}$ denote the series

[TABLE]

Note that if either $a$ or $b$ is negative then the series is finite.

Now, changing the index of the series on the left hand side of the desired identity to $k=i-1$ , we let $C_{k}$ be the $k^{\textrm{th}}$ term. Expanding factorials and cancellation gives

[TABLE]

This means that the left hand side of the desired identity can be written as

[TABLE]

The Chu-Vandermonde Identity (Roy, 87, Equation (2.7)) gives the desired identity:

[TABLE]

5 The number of equations of $\operatorname{\overline{{M}}}_{0,n}$ in $\operatorname{\overline{{M}}}_{0,n-1}\times\mathbb{P}^{n-3}$

We recall cohomological tools developed in KT (09). Working in the ideal defining $\operatorname{\overline{{M}}}_{0,n}$ as a subscheme of $\operatorname{\overline{{M}}}_{0,n-1}\times\mathbb{P}^{n-3}$ , these tools allow us to realize the number of equations of a given bidegree as the dimension of the space of global sections of a certain sheaf on $\operatorname{\overline{{M}}}_{0,n-1}$ . We apply this to the case $n=5$ to prove that $J_{5}$ contains all polynomials in $I_{5}$ of degree $(d,d)$ . We do the same for $J_{6}$ and $I_{6}$ in the next section.

Let $\sigma:U\to\operatorname{\overline{{M}}}_{0,n}$ be the universal curve over $\operatorname{\overline{{M}}}_{0,n}$ and $\omega$ the relative dualizing sheaf. The $\kappa$ * class* on $\operatorname{\overline{{M}}}_{0,n}$ is the pushforward of the first Chern class of $\omega$ , i.e. $\kappa=\sigma_{*}(c_{1}(\omega))$ . Let $K_{\operatorname{\overline{{M}}}_{0,n}}$ be the canonical class on $\operatorname{\overline{{M}}}_{0,n}$ and $\delta_{I}$ the classes of the boundary divisors. In KT (09), Keel and Tevelev prove that $\kappa\sim K_{\operatorname{\overline{{M}}}_{0,n}}+\sum{\delta_{I}}$ , that $\kappa$ is very ample, and that the composition of $\phi$ with the Segre embedding $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}\hookrightarrow\mathbb{P}^{(n-2)!-1}$ is the embedding of $\operatorname{\overline{{M}}}_{0,n}$ via the $\kappa$ class. Proposition 2 implies:

Corollary 3

The ideal of the embedding of $\operatorname{\overline{{M}}}_{0,5}$ into $\mathbb{P}^{5}$ via the $\kappa$ class is generated by the five quadrics in (1).

Proof

The first two equations are given by first multiplying the equation of Proposition 2 by $x_{0}$ and $x_{1}$ to obtain equations homogeneous of the same degree in $x_{i}$ and $y_{j}$ , then mapping into $\mathbb{P}^{5}$ by the Segre embedding. The final three are the Segre relations. By KT (09), the resulting map from $\operatorname{\overline{{M}}}_{0,5}\to\mathbb{P}^{5}$ is given by the $\kappa$ class. ∎

We will prove that for $n=5,6$ , the ideals generated by polynomials in $I_{n}$ of degree $(d,d,\ldots,d)$ are generated by polynomials in $J_{n}$ . The following theorem tells us that this is enough to understand the ideal of $\Phi(\operatorname{\overline{{M}}}_{0,n})$ in $\mathbb{P}^{(n-2)!-1}$ via the $\kappa$ class.

Theorem 5.1

KT (09)** The ideal $A_{n}$ that defines $\Phi(\operatorname{\overline{{M}}}_{0,n})$ as a subscheme of $\mathbb{P}^{(n-2)!-1}$ is generated by quadrics. Equivalently, let $\tilde{I}_{n}$ be the ideal generated by all polynomials of degree $(d,d,\ldots,d)$ contained in the ideal $I_{n}$ in the Cox ring of $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{n-3}$ . Then $\tilde{I}_{n}$ is generated by polynomials of degree $(2,\ldots,2)$ .

Let $V_{\psi_{n}}$ be the vector bundle on $\operatorname{\overline{{M}}}_{0,n}$ defined by the exact sequence

[TABLE]

and consider the map $\Phi=(\pi_{n},\psi_{n}):\operatorname{\overline{{M}}}_{0,n}\to\operatorname{\overline{{M}}}_{0,n-1}\times\mathbb{P}^{n-3}$ . By (KT, 09, Lemma 4.1), we have the following resolution of the structure sheaf of $\Phi(\operatorname{\overline{{M}}}_{0,n})$ in $\operatorname{\overline{{M}}}_{0,n-1}\times\mathbb{P}^{n-3}$ :

[TABLE]

where $\mathcal{M}_{n}^{p}=R^{1}\pi_{n*}(\wedge^{p+1}V_{\psi_{n}})$ .

To gain control of the sheaves in (6), we have

Theorem 5.2

[KT09, Theorem 4.3] There exists a vector bundle $Q$ on $\operatorname{\overline{{M}}}_{0,n}$ and exact sequences

[TABLE]

[TABLE]

Remark 2

The vector bundle $Q$ is defined explicitly in KT (09).

Thus, by tensoring (6) with the correct line bundles one can compute the expected number of equations for $\operatorname{\overline{{M}}}_{0,n}\subset\operatorname{\overline{{M}}}_{0,n}\times\mathbb{P}^{n-3}$ in particular degrees.

Lemma 7

For all integers $n\geq 5$ and $a>0$ , the ideal $I_{n}$ defining $\operatorname{\overline{{M}}}_{0,n}$ as a subscheme of $\operatorname{\overline{{M}}}_{0,n-1}\times\mathbb{P}^{n-3}$ contains exactly $h^{0}(\operatorname{\overline{{M}}}_{0,n-1},\mathcal{M}^{1}_{n-1}\otimes\kappa_{n-1}^{\otimes a})$ linearly independent equations of bidegree $(a,2)$ . Additionally, we have the short exact sequence

[TABLE]

[TABLE]

Proof

We use the resolution of the structure sheaf of $\Phi_{*}\mathcal{O}_{\operatorname{\overline{{M}}}_{0,n}}$ given by the exact sequence (6). Tensoring this with $\kappa_{n-1}^{\otimes a}\boxtimes\mathcal{O}(2)$ , we obtain the following exact sequence on $\operatorname{\overline{{M}}}_{0,n-1}\times\mathbb{P}^{n-3}$ :

[TABLE]

[TABLE]

[TABLE]

By the Künneth Formula, and since $\mathcal{O}(k)$ is acyclic for $1-n\leq k\leq-1$ , we have

[TABLE]

for all $1-n\leq k\leq-1$ and $i\geq 0$ . Moreover, by Lemma 6.5 of KT (09), we have

[TABLE]

Thus, we obtain a short exact sequence in cohomology:

[TABLE]

[TABLE]

In particular, the number of equations of bidegree $(a,2)$ in the ideal defining $\operatorname{\overline{{M}}}_{0,n}$ as a subvariety of $\operatorname{\overline{{M}}}_{0,n-1}\times\mathbb{P}^{n-3}$ is given by the dimension of the kernel of $\tau$ , which is $H^{0}(\operatorname{\overline{{M}}}_{0,n-1}\times\mathbb{P}^{n-3},(\mathcal{M}^{1}_{n-1}\otimes\kappa_{n-1}^{\otimes a})\boxtimes\mathcal{O})$ . By the Künneth Formula, this vector space is isomorphic to $H^{0}(\operatorname{\overline{{M}}}_{0,n-1},\mathcal{M}^{1}_{n-1}\otimes\kappa_{n-1}^{\otimes a})$ .

Using the short exact sequences (7) and (8), and noting that $\mathcal{M}^{0}_{n}=0$ for all $n$ , we have the short exact sequence

[TABLE]

We tensor with $\kappa_{n-1}^{\otimes a}$ . Then $H^{i}(\operatorname{\overline{{M}}}_{0,n-1},\pi_{n-1}^{*}\mathcal{M}^{1}_{n-2}\otimes\kappa_{n-1}^{\otimes a})=0$ for $i>0$ , by Lemma 6.5 in KT (09), and so taking cohomology gives the short exact sequence. ∎

Example 6

For a reality check, recall from Proposition 2 that $I_{5}$ is generated by one equation of degree $(1,2)$ , and Lemma 4 gives one equation satisfied by $\operatorname{\overline{{M}}}_{0,5}$ in $\mathbb{P}^{1}\times\mathbb{P}^{2}$ . On the other hand, Lemma 7 tells us that $\operatorname{\overline{{M}}}_{0,5}$ has $h^{0}(\operatorname{\overline{{M}}}_{0,4},\mathcal{M}^{1}_{4}\otimes\kappa_{4}^{\otimes a})$ linearly independent equations of degree $(a,2)$ .

Theorem 5.3

We have $h^{0}(\operatorname{\overline{{M}}}_{0,4},\mathcal{M}^{1}_{4}\otimes\kappa_{4})=1$ and $h^{0}(\operatorname{\overline{{M}}}_{0,4},\mathcal{M}^{1}_{4}\otimes\kappa_{4}^{\otimes 2})=2$ .

Proof

Under the isomorphism $\operatorname{\overline{{M}}}_{0,4}\simeq\mathbb{P}^{1}$ , we have $\kappa_{4}=\mathcal{O}_{\mathbb{P}^{1}}(1)$ . We show that on $\operatorname{\overline{{M}}}_{0,4}$ , we have $\mathcal{M}^{1}_{4}=\psi_{4}^{-1}=\mathcal{O}_{\mathbb{P}^{1}}(1)$ . Since $\mathcal{M}^{1}_{3}=0$ , the exact sequence (7) becomes

[TABLE]

and tells us that $\mathcal{M}^{1}_{4}\simeq Q$ . Since $\mathcal{M}^{0}_{n}=0$ for all $n$ , the sequence (8) becomes

[TABLE]

and therefore $\mathcal{M}^{1}_{4}\simeq Q\simeq V_{\psi_{4}}$ . We can determine the bundle $V_{\psi_{4}}$ by analyzing the exact sequence (5). On $\operatorname{\overline{{M}}}_{0,4}$ , this is

[TABLE]

Taking determinants gives the following equality of line bundles:

[TABLE]

In particular, $\mathcal{M}^{1}_{4}\simeq V_{\psi_{4}}$ is a line bundle dual to $\psi_{4}$ . Recall that $\operatorname{\overline{{M}}}_{0,4}\simeq\mathbb{P}^{1}$ and $\psi_{4}\simeq\mathcal{O}_{\mathbb{P}^{1}}(1)$ , so the dimension of $H^{0}(\operatorname{\overline{{M}}}_{0,4},\mathcal{M}_{4}^{1}\otimes\mathcal{O}_{\mathbb{P}^{1}}(2))$ is given by

[TABLE]

Similarly we compute the dimension of $H^{0}(\operatorname{\overline{{M}}}_{0,4},\mathcal{M}_{4}^{1}\otimes\mathcal{O}_{\mathbb{P}^{1}}(1))$ to be

[TABLE]

6 Equations for $\operatorname{\overline{{M}}}_{0,6}$

We apply the tools from the previous section to prove Theorem 6.1, which states that $\tilde{I}_{6}$ is generated by polynomials in $J_{6}$ .

Lemma 4 gives five polynomials satisfied by $\operatorname{\overline{{M}}}_{0,6}$ in $\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{P}^{3}$ :

[TABLE]

Let $J_{6}$ be the ideal in $\mathbb{C}[a_{0},a_{1},b_{0},b_{1},b_{2},c_{0},c_{1},c_{2},c_{3}]$ generated by $f_{1},\ldots,f_{5}$ , and let $I_{6}$ be the unique $B$ -saturated ideal defining $\operatorname{\overline{{M}}}_{0,6}$ scheme-theoretically, where

[TABLE]

Using Macaulay2, we verified that $I_{6}$ is prime, and is generated by $J_{6}$ and

[TABLE]

Proposition 3

The ideal $J_{6}$ is properly contained in the ideal $I_{6}$ , but the parts of homogeneous degrees $(2,2,2)$ of the ideals $J_{6}$ and $I_{6}$ coincide.

Proof

Let $I_{(i,j,k)}$ , respectively $J_{(i,j,k)}$ , be the vector space of polynomials of degree $(i,j,k)$ in $I_{6}$ , respectively $J_{6}$ . Multiplying the polynomials $f_{1},\ldots,f_{5}$ by all monomials of the correct degree gives a spanning set of $J_{(i,j,k)}$ . Computing the dimension of $J_{(i,j,k)}$ involves determining which of the resulting polynomials are redundant. We used Macaulay2 to show that $\dim J_{(1,1,2)}=9<\dim I_{(1,1,2)}=10$ and $\dim J_{(2,2,2)}=\dim I_{(2,2,2)}=55$ . ∎

Remark 3

As we will see, the second part of Proposition 3 implies that the homogeneous parts of the ideals $I_{6}$ and $J_{6}$ coincide. Corollary 4 shows that these ideals contain the correct number of homogeneous equations of degree $(2,2,2)$ .

Theorem 6.1

Let $\tilde{I}_{6}$ be the ideal generated by the polynomials of degree $(d,d,d)$ in $I_{6}$ . Then $\tilde{I}_{6}$ is generated by the polynomials of degree $(2,2,2)$ contained in $J_{6}$ . Equivalently, the embedding $\Phi(\operatorname{\overline{{M}}}_{0,n})$ in $\mathbb{P}^{23}$ defined by the $\kappa$ class is generated by the homogeneous polynomials of degree $(2,2,2)$ in $J_{6}$ and the Segre relations.

The proof of Theorem 6.1 requires the following two lemmas.

Lemma 8

We have $h^{0}(\operatorname{\overline{{M}}}_{0,5},V_{\psi_{4}}\otimes\kappa_{5}^{\otimes 2})=24$ and $h^{0}(\operatorname{\overline{{M}}}_{0,5},V_{\psi_{4}}\otimes\kappa_{5})=11$ .

Proof

We have on $\operatorname{\overline{{M}}}_{0,5}$ the short exact sequence

[TABLE]

We tensor with $\kappa_{5}^{\otimes 2}$ . Noting that $H^{i}(\operatorname{\overline{{M}}}_{0,5},V_{\psi_{4}}\otimes\kappa_{5}^{\otimes 2})=0$ for $i>0$ by (KT, 09, Lemma 6.5), the long exact sequence in cohomology gives the short exact sequence

[TABLE]

We shall determine the dimensions of the middle term and the last term.

For the middle term, we note first that any global section of $\mathbb{C}^{3}\otimes\kappa_{5}^{\otimes 2}$ is of the form $\alpha\otimes\beta$ where $\alpha\in\mathbb{C}^{3}$ and $\beta\in H^{0}(\operatorname{\overline{{M}}}_{0,5},\kappa_{5}^{\otimes 2})$ . Using that $\operatorname{\overline{{M}}}_{0,5}\simeq\textrm{Bl}_{q_{1},\ldots,q_{4}}\mathbb{P}^{2}$ , we let $\sigma:\textrm{Bl}_{q_{1},\ldots,q_{4}}\mathbb{P}^{2}\to\mathbb{P}^{2}$ be the blowup, $E_{1},\ldots,E_{4}$ the exceptional divisors, and $L_{i,j}$ the proper transform of the line passing through $q_{i}$ and $q_{j}$ on $\mathbb{P}^{2}$ . The $\kappa$ class is given by $\kappa_{5}=K_{\operatorname{\overline{{M}}}_{0,5}}+\sum\delta_{I}$ , where the sum is taken over all boundary divisors $\delta_{I}$ of $\operatorname{\overline{{M}}}_{0,5}$ , so we have the linear equivalence

[TABLE]

Since $L_{i,j}\sim\sigma^{*}{H}-E_{i}-E_{j}$ , where $H$ is the class of a hyperplane section on $\mathbb{P}^{2}$ , we can write $\kappa_{5}^{\otimes 2}$ as $\,(\kappa_{5})^{\otimes 2}\sim\sigma^{*}(6H)-2\sum_{i=1}^{4}E_{i}$ . In particular, this gives

[TABLE]

where $\mathcal{I}_{q}$ denotes the skyscraper sheaf at $q$ . By explicitly writing out equations, one can check that the conditions that a curve $C$ of degree $6$ on $\mathbb{P}^{2}$ attain nodes at four points in general position are linearly independent. Thus, this latter vector space has dimension $\binom{8}{2}-12=16$ , and therefore $h^{0}(\operatorname{\overline{{M}}}_{0,5},\mathbb{C}^{3}\otimes\kappa_{5}^{\otimes 2})=3\cdot 16=48.$

Repeating the above with $\kappa_{5}^{\otimes 2}$ replaced by $\kappa_{5}$ , we have $h^{0}(\operatorname{\overline{{M}}}_{0,5},\mathbb{C}^{3}\otimes\kappa_{5})=3\begin{pmatrix}\binom{5}{2}-4\end{pmatrix}=18.$

Next, since $\psi_{4}\sim\delta_{1,2}+\delta_{3,5}+\delta_{4,5}\sim H$ , we have $\psi_{4}\otimes\kappa_{5}^{\otimes 2}\,\sim\,\sigma^{*}(7H)-2(E_{1}+E_{2}+E_{3}+E_{4})$ . With this, we compute the dimension of $H^{0}(\operatorname{\overline{{M}}}_{0,5},\psi_{4}\otimes\kappa_{5}^{\otimes 2})$ to be

[TABLE]

Finally, we repeat the above computation with $\kappa_{5}^{\otimes 2}$ replaced by $\kappa_{5}$ , and we find

[TABLE]

Lemma 9

We have $\,h^{0}(\operatorname{\overline{{M}}}_{0,5},\pi^{*}\mathcal{M}^{1}_{4}\otimes\kappa_{5}^{\otimes 2})=11\,$ and $\,h^{0}(\operatorname{\overline{{M}}}_{0,5},\pi^{*}\mathcal{M}^{1}_{4}\otimes\kappa_{5})=3$ .

Proof

We note first that on $\operatorname{\overline{{M}}}_{0,4}\simeq\mathbb{P}^{1}$ , we have $\mathcal{M}^{1}_{4}=\psi_{4}^{-1}$ and $\psi_{4}=\mathcal{O}(-1)$ , so $\pi^{*}(\mathcal{M}^{1}_{4})=\pi^{*}(\psi_{4}^{-1})$ . Thus, $\,\pi^{*}(\mathcal{M}^{1}_{4})\,=\,\pi^{*}(\mathcal{O}(1))=-2H+\sum_{i=1}^{4}E_{i}$ . This allows us to write the class $\pi^{*}\mathcal{M}^{1}_{4}\otimes\kappa_{5}^{\otimes 2}$ as

[TABLE]

Using the push-pull formula, we can now compute the desired dimensions:

[TABLE]

[TABLE]

Applying Lemmas 8 and 9 to the short exact sequence of Lemma 7 gives

Corollary 4

The number of equations defining $\operatorname{\overline{{M}}}_{0,6}$ in the line bundle $\kappa_{5}^{\otimes 2}\boxtimes\mathcal{O}(2)$ on $\operatorname{\overline{{M}}}_{0,5}\times\mathbb{P}^{3}$ is $35$ . Similarly, for the line bundle $\kappa_{5}\boxtimes\mathcal{O}(2)$ , the number is $10$ .

Proof

For $n=6$ and $a=1,2$ , Lemma 7 gives the short exact sequences

[TABLE]

[TABLE]

The result follows from these, and Lemmas 8 and 9. ∎

Proof (of Theorem 6.1)

There are no equations in $I_{5}$ of degree $(1,1)$ , so the ideal generated by equations of degree $(1,1,2)$ in $I_{6}$ has number of linearly independent polynomials coinciding with the number of linearly independent polynomials defining $\operatorname{\overline{{M}}}_{0,6}$ in the line bundle $\kappa_{5}\boxtimes\mathcal{O}(2)$ on $\operatorname{\overline{{M}}}_{0,5}\times\mathbb{P}^{3}$ .

Since there are two equations in $I_{5}$ of degree $(2,2)$ in the Cox ring of $\mathbb{P}^{1}\times\mathbb{P}^{2}$ , both of which must be homogenized, we see that $I_{6}$ contains 20 linearly independent polynomials of degree $(2,2,2)$ in the Cox ring of $\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{P}^{3}$ . By Corollary 4, the number of linearly independent polynomials of degree $(2,2,2)$ is $h^{0}(\operatorname{\overline{{M}}}_{0,5},\mathcal{M}^{1}_{5}\otimes\kappa_{5}^{\otimes 2})+20=55$ . The number of linearly independent polynomials in $I_{6}$ of degree $(1,1,2)$ equals $10$ . Together with Proposition 3, this completes the proof.∎

7 Appendix

We include the Macaulay2 code used to verify Conjecture 1.

R = QQ[a0,a1,b0,b1,b2,c0,c1,c2,c3,d0,d1,d2,d3,d4, Ψe0,e1,e2,e3,e4,e5,f0,f1,f2,f3,f4,f5,f6];

The rows of the following matrix are coordinates on $\mathbb{P}^{1},\mathbb{P}^{2},\mathbb{P}^{3},\mathbb{P}^{4},\mathbb{P}^{5},\mathbb{P}^{6}$

M=matrix{{0,0,0,0,0,0,0},{a0,a1,0,0,0,0,0}, Ψ{b0,b1,b2,0,0,0,0},{c0,c1,c2,c3,0,0,0}, Ψ{d0,d1,d2,d3,d4,0,0},{e0,e1,e2,e3,e4,e5,0}, Ψ{f0,f1,f2,f3,f4,f5,f6}};

M05 = {{2,2}}; M06 = {{2,2}, {3,2},{3,3}}; M07 = {{2,2}, {3,2},{3,3}, {4,2},{4,3},{4,4}}; M08 = {{2,2}, {3,2},{3,3}, {4,2},{4,3},{4,4},{5,2}, Ψ{5,3},{5,4},{5,5}}; M09 = {{2,2}, {3,2},{3,3}, {4,2},{4,3},{4,4}, {5,2}, Ψ{5,3},{5,4},{5,5}, {6,2},{6,3},{6,4},{6,5},{6,6}};

Select your desired $n$ here:

L = M07;

Lemma 4 involves the $2\times 2$ -minors of the matrices

Q=apply(L,l->{submatrix(M,{l_1-1,l_0},0..l_1-1), ΨΨΨM_(l_1)_(l_0)}) S=apply(Q,T-> matrix{apply(entries transpose T_0, ΨΨΨx->x_0*(x_1-T_1)),(entries T_0)_1})

We form the ideal $J=J_{n}$ of all such $2\times 2$ -minors, and compute the prime ideal $I$ by saturation.

J = sum apply(S,N -> minors(2,N)); J = saturate(J,ideal(a0,a1)); J = saturate(J,ideal(b0,b1,b2)); J = saturate(J,ideal(c0,c1,c2,c3)); J = saturate(J,ideal(d0,d1,d2,d3,d4)); J = saturate(J,ideal(e0,e1,e2,e3,e4,e5)); I = saturate(J,ideal(f0,f1,f2,f3,f4,f5,f6));

The following are used to determine whether the dimension of $I$ is correct, as well as compute the degree of $I$ , and the minimal number of generators in each degree.

codim I, degree I, betti mingens I

We finally note that the initial ideal is square-free and Cohen-Macaulay:

M = monomialIdeal leadTerm I; betti mingens M

Acknowledgements.

This article was initiated during the Apprenticeship Weeks (22 August-2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. We thank Bernd Sturmfels for providing inspiration and feedback on multiple drafts. We are also grateful to Christine Berkesch Zamaere, Renzo Cavalieri, Diane Maclagan, Steffen Marcus, Vic Reiner, and Jenia Tevelev for many helpful discussions. The second author was partially supported by a scholarship from the Clay Math Institute.

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