On the cohomology of moduli spaces of (weighted) stable rational curves

Abstract

We give a recursive algorithm for computing the character of the cohomology of the moduli space ${\overline{M}}_{0,n}$ of stable $n$-pointed genus zero curves as a representation of the symmetric group $\mathbb{S }_n$ on $n$ letters. Using the algorithm we can show a formula for the maximum length of this character. Our main tool is connected to the moduli spaces of weighted stable curves introduced by Hassett.

Appendices

Appendix A: Cohomology of the blow-up

First, we recall the fact that the cohomology of the blow-up $\widetilde{M}$ of a smooth projective variety $M$ along a smooth subvariety $Z$ of codimension $l$ is given by

$$\begin{aligned} H^k(\widetilde{M})\cong H^k(M)\oplus \bigoplus _{i=1}^{l-1} \left( H^{k-2i}(Z)\otimes H^{2i}(\mathbb P ^{l-1})\right) ~, \end{aligned}$$

(6.1)

see e.g. [12, Theorem 7.31].

Let $X$ be a smooth projective variety and $Y=Y_1 \cup \cdots \cup Y_n$ be the union of smooth subvarieties of $X$. Let $\widetilde{X}$ be the blow-up of $X$ along $Y$. We assume that any non-empty intersection of irreducible components of $Y$ is transversal. Then it is known that $\widetilde{X}$ is obtained by a sequence of smooth blow-ups along the proper transforms of the irreducible components of $Y$ in any order, see e.g. [7, Proposition 2.10]. We further assume that $\mathrm{codim}_X Y_i=l$ for any $i$. The aim of this appendix is to give a formula for $H^*(\widetilde{X})$. For a subset $I$ of $\{1, \ldots , n\}$, we set $Y_I:=\cap _{i\in I} Y_i$, which is a smooth subvariety of $X$ by the assumption of transversality.

Proposition 6.1

Under the above assumptions, we have

$$\begin{aligned} H^*(\widetilde{X}) \cong H^*(X) \oplus \bigoplus _{I \subset \{1,\ldots ,n\}} \left( H^*(Y_I) \otimes \left( H^+(\mathbb P ^{l-1}) \right) ^{\otimes |I|}\right) ~, \end{aligned}$$

(6.2)

where $I$ runs over all subsets of $\{1,\ldots , n\}$ for which $Y_I \ne \emptyset$.

Proof

Let

$$\begin{aligned} X_{n} \overset{\pi _n}{\longrightarrow } X_{n-1} \rightarrow \cdots \rightarrow X_1 \overset{\pi _1}{\longrightarrow } X_0 \end{aligned}$$

be the sequence of blow-ups which we define inductively as follows.

• (i) Let $\pi _1: X_1\rightarrow X_0:=X$ be the blow-up along $Y_1$.

• (ii) For $i\ge 2$, we put $\pi _{i}: X_{i} \rightarrow X_{i-1}$ to be the blow-up along the proper transform of $Y_i$.

Note that $X_n$ is isomorphic to $\widetilde{X}$. For $1\le j \le i-1$, we denote by $Y_{i,j}$ the proper transform of $Y_i$ under $\pi _{j}\circ \cdots \circ \pi _1: X_{j} \rightarrow X_{0}$. In this notation, $\pi _i: X_i\rightarrow X_{i-1}$ is the blow-up along ${Y}_{i, i-1}$. Then it follows from (6.1) that

$$\begin{aligned} H^*(X_i) \cong H^*(X_{i-1}) \oplus \left( H^*({Y}_{i,i-1}) \otimes H^+(\mathbb P ^{l-1}) \right) , \end{aligned}$$

(6.3)

as graded vector spaces. The Eq. (6.3) together with (6.4) given in Lemma 6.2 below implies (6.2).$\square$

Lemma 6.2

Under the same assumptions and notation as above, we have

$$\begin{aligned} H^*({Y}_{i,i-1})\cong H^*(Y_i) \oplus \bigoplus _{I\subset \{1,\ldots ,i-1\}} \left( H^*(Y_i \cap Y_I)\otimes \left( H^+ (\mathbb P ^{l-1}) \right) ^{\otimes |I|} \right) , \end{aligned}$$

(6.4)

where $I$ runs over all the subsets of $\{1,\ldots , i-1\}$ for which $Y_i\cap Y_I \ne \emptyset$.

Proof

Note that the proper transform $Y_{i,i-1}$ of $Y_i$ is obtained by the sequence of blow-ups

$$\begin{aligned} Y_{i,i-1}\overset{\pi _{i-1}}{\longrightarrow } Y_{i,i-2}\rightarrow \cdots \rightarrow Y_{i,1} \overset{\pi _1}{\longrightarrow } Y_{i,0}:=Y_i~, \end{aligned}$$

and that the center of the blow-up $\pi _j: Y_{i,j}\rightarrow Y_{i,j-1}$ is $Y_{i,j-1}\cap Y_{j,j-1}$. In general, for $I\subset \{1,\ldots , n\}$ and $j<\min \{i\in I\}$ such that $Y_{I,j}:=\cap _{i\in I}Y_{i,j}\ne \emptyset , Y_{I,j}$ is the blow-up of $Y_{I,j-1}$ along $Y_{I,j-1}\cap Y_{j,j-1}$. Using this structure and (6.1) recursively, we obtain

$$\begin{aligned} H^*({Y}_{i,j})\cong H^*(Y_i) \oplus \bigoplus _{I\subset \{1,\ldots ,j\}} \left( H^*(Y_i \cap Y_I)\otimes \left( H^+(\mathbb P ^{l-1}) \right) ^{\otimes |I|} \right) , \end{aligned}$$

(6.5)

where $I$ runs over all the subsets of $\{1,\ldots , j\}$ for which $Y_I\cap Y_i \ne \emptyset$. The desired formula (6.4) is the case $j=i-1$ in (6.5).$\square$

Appendix B: Characters of representations of symmetric groups

Let $\mathbb{S }_n$ be the symmetric group on $n$ letters. Let $\Lambda :=\underset{\longleftarrow }{\lim }~\mathbb Z [x_1, \ldots , x_n]^{\mathbb{S }_n}$ be the ring of symmetric functions. It is well known that $\Lambda \otimes \mathbb Q =\mathbb Q [p_1, p_2, \ldots ]$ where $p_n$ are the power sums. We denote by $\mathcal{P }(n)$ the set of partitions of $n$, and for $\lambda =(\lambda _1,\ldots ,\lambda _{l(\lambda )}) \in \mathcal{P }(n)$ we set $p_{\lambda }:=\prod \nolimits _{i=1}^{l(\lambda )} p_{\lambda _i}$. We also set $\Lambda ^{x, y}:=\Lambda ^x\otimes \Lambda ^y$, where $\Lambda ^x$ and $\Lambda ^y$ are the ring of symmetric functions in $x=(x_1, x_2, \ldots )$ and $y=(y_1, y_2, \ldots )$, respectively.

For a representation $V$ of $\mathbb{S }_n$, we define

$$\begin{aligned} \mathrm{ch}_n(V) :=\frac{1}{n!} \sum _{w \in \mathbb{S }_n} \mathrm{Tr}_V(w) p_{\rho (w)} \in \Lambda , \end{aligned}$$

where $\rho (w)\in \mathcal{P }(n)$ is the cycle type of $w\in \mathbb{S }_n$. Similarly we define, for an $\mathbb{S }_k \times \mathbb{S }_{n-k}$ representation $V$,

$$\begin{aligned} \mathrm{ch}_{k, n-k}(V) :=\frac{1}{k! }\frac{1}{(n-k)!} \sum _{(u,v) \in \mathbb{S }_k \times \mathbb{S }_{n-k}} \mathrm{Tr}_V\left( (u,v)\right) p_{\rho (u)}^x \, p_{\rho (v)}^y \in \Lambda ^{x, y}, \end{aligned}$$

where $p_n^x$ and $p_n^y$ are the power sums in the variable $x$ and $y$, respectively.

If $V$ and $W$ are representations of $\mathbb{S }_n$ we put

$$\begin{aligned} \mathrm{ch}_{n}(V) * \mathrm{ch}_{n}(W):=\mathrm{ch}_{n}(V \otimes W). \end{aligned}$$

For any $\lambda \in \mathcal{P }(k)$, if we put $m_j(\lambda ):=\#\{ i \mid \lambda _i =j\}$ and

$$\begin{aligned} \frac{\partial }{\partial p^x_{\lambda }}:=\left( \prod _{i=1}^{k}\frac{1}{m_i(\lambda )!} \right) \frac{\partial }{\partial p^x_{\lambda _1}} \frac{\partial }{\partial p^x_{\lambda _2}} \cdots \frac{\partial }{\partial p^x_{\lambda _{l(\lambda )}}}, \end{aligned}$$

then

$$\begin{aligned} \mathrm{ch}_{k,n-k}\left( \mathrm{Res}^{\mathbb{S }_{n}}_{\mathbb{S }_{k} \times \mathbb{S }_{n-k}}(V)\right) =\sum _{\lambda \in \mathcal{P }(n-k)} \left( \frac{\partial }{\partial p^x_{\lambda }} \mathrm{ch}_{n}(V) \right) \, p^y_{\lambda }. \end{aligned}$$

If $V_i$ are representations of $\mathbb{S }_{n_i}$ for $1 \le i \le k$ then

$$\begin{aligned} \mathrm{ch}_{\sum \nolimits _{i=1}^k n_i} \left( \mathrm{Ind}_{\mathbb{S }_{n_1} \times \cdots \times \mathbb{S }_{n_k}}^{\mathbb{S }_{\sum \nolimits _{i=1}^k n_i}} (V_1 \boxtimes V_2 \boxtimes \cdots \boxtimes V_k) \right) = \prod _{i=1}^k \mathrm{ch}_{n_i}(V_i). \end{aligned}$$

If $\circ$ denotes plethysm between symmetric functions, and $\sim$ denotes the wreath product, that is, $\mathbb{S }_{n_1} \sim \, \mathbb{S }_{n_2}:=\mathbb{S }_{n_1} \ltimes (\mathbb{S }_{n_2})^{n_1}$ where $\mathbb{S }_{n_1}$ acts on $(\mathbb{S }_{n_2})^{n_1}$ by permutation, then

$$\begin{aligned} \mathrm{ch}_{n_1 n_2}\left( \mathrm{Ind}_{\mathbb{S }_{n_1} \sim \, \mathbb{S }_{n_2}}^{\mathbb{S }_{n_1 n_2}} (V_1 \boxtimes \underbrace{ V_2 \boxtimes \cdots \boxtimes V_2 }_{n_1} \,) \right) = \mathrm{ch}_{n_1}(V_1) \circ \mathrm{ch}_{n_2}(V_2), \end{aligned}$$

see [9, Appendix A, p. 158].

Recall finally that irreducible representations of $\mathbb{S }_n$ are indexed by $\mathcal{P }(n)$. For $\lambda \in \mathcal{P }(n)$, let $V_{\lambda }$ be the irreducible representation corresponding to $\lambda$ and define the Schur function

$$\begin{aligned} s_{\lambda }:=\mathrm{ch}_n(V_{\lambda })\in \Lambda . \end{aligned}$$

It is well-known that $\{s_{\lambda }\}$, where $\lambda$ runs over all the partitions, is a $\mathbb Z$-basis of $\Lambda$.

Appendix C: An ordering of symmetric functions

For any partition $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _{l(\lambda )})$ we denote its dual partition by $\lambda ^{\prime }=(\lambda _1^{\prime },\lambda _2^{\prime },\ldots )$, where $\lambda _i^{\prime }:= |\{j:\lambda _j \ge i \}|$. If $\mu =(\mu _1,\mu _2,\ldots ,\mu _{l(\mu )})$ is another partition we define the partitions $\lambda +\mu$ as $(\lambda _1+\mu _1,\lambda _2+\mu _2,\ldots )$ and the partition $\lambda \cup \mu$ as the reordering of $(\lambda _1,\ldots ,\lambda _{l(\lambda )},\mu _1,\mu _2,\ldots ,\mu _{l(\mu )})$. Note that $\lambda \cup \mu =(\lambda ^{\prime }+\mu ^{\prime })^{\prime }$.

We introduce an ordering. For any partitions $\lambda$ and $\mu$ we say that $\lambda > \mu$ if there is a $k$ such that $\lambda _i^{\prime }=\mu _i^{\prime }$ for $1 \le i \le k-1$ and $\lambda _k^{\prime }> \mu _k^{\prime }$.

Definition 8.1

For any symmetric function $f=\sum \nolimits _{\lambda } a_{\lambda } s_{\lambda }$ we let $w(f)$ be the maximal partition $\lambda$ (w.r.t. $>$) such that $a_\lambda \ne 0$.

For any partition $\lambda$, we put $h_{\lambda }:=\prod \nolimits _{i=1}^{l(\lambda )} s_{(\lambda _i)}$ and $e_{\lambda }:=\prod \nolimits _{i=1}^{l(\lambda )} s_{(1^{\lambda _i})}$. The following is well known.

Lemma 8.2

There are integers $a_{\lambda ,\mu }$ and $b_{\lambda ,\mu }$ such that

$$\begin{aligned} s_{\lambda }=h_{\lambda }+\sum _{\mu < \lambda } a_{\lambda ,\mu }h_{\mu }=e_{\lambda ^{\prime }}+\sum _{\mu < \lambda ^{\prime }} b_{\lambda ,\mu }e_{\mu }. \end{aligned}$$

Lemma 8.3

For any symmetric functions $f$ and $g$ we have, $w(fg)=w(f) \cup w(g).$

Proof

Since $h_{\mu } h_{\nu }=h_{\mu \cup \nu }$, it follows from Lemma 8.2 that there are integers $c_{\lambda }$ such that $fg=c_{w(f) \cup w(g)}h_{w(f) \cup w(g)}+\sum \nolimits _{\lambda < w(f) \cup w(g)} c_{\lambda } h_{\lambda }$ with $c_{w(f) \cup w(g)}\ne 0$.$\square$

Due to the lack of a suitable reference we will show here how $w$ behaves with respect to plethysm between symmetric functions.

Lemma 8.4

[9, p. 158] For any symmetric functions $f$, $g$ and $h$ we have,

$$\begin{aligned} (f \circ h) (g \circ h)=(f \, g) \circ h. \end{aligned}$$

We denote by $(\cdot \,| \cdot )$ the standard inner product on $\Lambda$ for which Schur functions are orthonormal.

Proposition 8.5

[10, Theorem I, IA] Say that $s_{(k)} \circ s_{(m)}=\sum \nolimits _{\lambda } a_{\lambda } s_{\lambda }$, then, for any $0 \le i \le k$,

$$\begin{aligned} (s_{(1^i)} \circ s_{(m-1)}) \, (s_{(k-i)} \circ s_{(m)}) = \sum _{\lambda } \sum _{\nu } a_{\lambda } \left( s_{(1^i)}s_{\nu } | s_{\lambda }\right) s_{\nu }. \end{aligned}$$

(8.1)

Similarly, say that $s_{(1^k)} \circ s_{(m)}=\sum \nolimits _{\lambda } b_{\lambda } s_{\lambda }$, then, for any $0 \le i \le k$,

$$\begin{aligned} (s_{(i)} \circ s_{(m-1)}) \, (s_{(1^{k-i})} \circ s_{(m)}) = \sum _{\lambda } \sum _{\nu } b_{\lambda } \left( s_{(1^i)}s_{\nu } | s_{\lambda }\right) s_{\nu }. \end{aligned}$$

(8.2)

Proposition 8.6

For any $m \ge 1$ and partition $\mu$ of $k$ we have

$$\begin{aligned} w(s_{\mu } \circ s_{(m)})= {\left\{ \begin{array}{ll} ({(m-1)}^k) + \mu \quad \,\text{ if }\quad m\, \text{ odd } \\ ({(m-1)}^k) + \mu ^{\prime } \quad \,\text{ if }\quad m\,\text{ even }. \end{array}\right. } \end{aligned}$$

(8.3)

Proof

We first recall that $l(s_{\mu } \circ s_{(m)}) \le |\mu |$, see [9, Example 9, p. 140]. Let us then continue by proving Eq. (8.3) for $\mu =(k)$ and $\mu =(1^k)$ by induction on $m$. The equation clearly holds for $m=1$. If $l(s_{\nu })=k$ and $l(s_{\lambda }) \le k$ then $(s_{(1^k)}s_{\nu }|s_{\lambda }) \ne 0$ implies that $\lambda =\nu + (1^k)$. Moreover, if $l(s_{\eta })<k$, $l(s_{\lambda }) \le k$ and $(s_{(1^k)}s_{\eta }|s_{\lambda }) \ne 0$ then $\lambda < \nu + (1^k)$ for any $\nu$ such that $l(\nu )=k$. Therefore, taking $i=k$ in Eq. (8.1) and applying the fact that $l(s_{(k)} \circ s_{(m)}) \le k$ we find by looking at the term with $\lambda =w(s_{(k)}\circ s_{(m)})$ on the right hand side of formula (8.1) that $w(s_{(k)} \circ s_{(m)})=w(s_{(1^k)} \circ s_{(m-1)}) + (1^k)$. Similarly, using Eq. (8.2) we find by induction on $m$ that $w(s_{(1^k)} \circ s_{(m)})=w(s_{(k)} \circ s_{(m-1)}) + (1^k)$. This completes the induction.

Finally, let us prove the statement for any $\mu$ and $m$. From Lemma 8.4 and from the formula for $w(s_{(k)} \circ s_{(m)})$ it follows, for $m$ odd, that

$$\begin{aligned} w(h_{\mu } \circ s_{(m)})= \bigcup _i w(s_{(\mu _i)} \circ s_{(m)})={((m-1)}^k) + \mu \end{aligned}$$

and similarly from the formula for $w(s_{(1^k)} \circ s_{(m)})$, for $m$ even, that

$$\begin{aligned} w(e_{\mu } \circ s_{(m)})= \bigcup _i w(s_{(1^{\mu _i})} \circ s_{(m)})={((m-1)}^k) + \mu . \end{aligned}$$

Using Lemma 8.2 it then follows, for $m$ odd, that

$$\begin{aligned} w(s_{\mu } \circ s_{(m)})=w \left( (h_{\mu }+\sum _{\nu < \mu } a_{\nu }h_{\nu }) \circ s_{(m)} \right) =w(h_{\mu } \circ s_{(m)}) \end{aligned}$$

and, for $m$ even, that

$$\begin{aligned} w(s_{\mu } \circ s_{(m)})=w \left( (e_{\mu ^{\prime }}+\sum _{\nu < \mu ^{\prime }} b_{\nu } e_{\nu }) \circ s_{(m)} \right) =w(e_{\mu ^{\prime }} \circ s_{(m)}). \end{aligned}$$

$\square$