Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal M}_{1,n}$

Introduction

The moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$ marked points is one of the most important objects in algebraic geometry [9, 15]. It parametrizes connected projective curves $C$ of arithmetic genus $g$ with at worst nodal singularities together with $n$ ordered smooth marked points $p_1,\cdots,p_n$ in $C$ such that

(1) $p_i$ are all distinct and

(2) $\omega_C\left(\sum_i p_i\right)$ is ample.

It is a smooth proper Deligne-Mumford stack with projective coarse moduli space. It plays a fundamental role in modern enumerative geometry and serves as a model for moduli spaces of higher dimensional varieties. For over half a century, much effort has been exerted to understand the geometry and topology of $\overline{\mathcal{M}}_{g,n}$ and there has been a constant stream of new discoveries even until now [1, 2, 4, 6, 7, 8, 13].

In this paper, we focus on the cohomology of $\overline{\mathcal{M}}_{1,n}$. In principle, the cohomology $H^*(\overline{\mathcal{M}}_{1,n})$ can be computed by [10] which gives a formula for the $S_n$-equivariant Hodge-Deligne polynomial of $\overline{\mathcal{M}}_{1,n}$. Hence we can compute the ordinary Hodge-Deligne polynomial as well as the $S_n$-action on the cohomology. Unfortunately, the formula is rather difficult to use for practical computation: We first need to compute the equivariant Hodge-Deligne polynomials of $\overline{\mathcal{M}}_{1,n}$, $\overline{\mathcal{M}}_{0,n}$ and $\overline{\mathcal{M}}_{0,n}$ for all $n$ and then expand a power series after applying the plethystic composition. As far as we know, computation for $\overline{\mathcal{M}}_{1,n}$ has been worked out only for $n$ up to 20 [3].

In contrast, the computation of the Poincaré polynomial of $\overline{\mathcal{M}}_{0,n}$ is much easier due to an explicit recursive formula [16] which can be proved by Keel’s blowup construction [14]. Therefore it seems reasonable to ask the following.

Question 1.1. Is there an explicit recursive formula for the Poincaré and Hodge-Deligne polynomials of $\overline{\mathcal{M}}_{1,n}$?

The purpose of this paper is to provide a positive answer to this question so that we get a new and effective way to compute the Hodge and Betti numbers of $\overline{\mathcal{M}}_{1,n}$.

Our computation of $H^*(\overline{\mathcal{M}}_{1,n})$ is achieved in two stages. We first reduce the computation to a simpler moduli space $\overline{\mathcal{M}}_{1,(0+)^n}$, where $0^+$ denotes a positive rational number sufficiently close to 0. We note that many of the strata in $\overline{\mathcal{M}}_{1,n}$ disappear and the computation becomes much easier if we allow all the marked points to coincide. This happens in the moduli space $\overline{\mathcal{M}}_{1,(0+)^n}$ of stable pointed curves with weight $0^+$ for each marked point [12]. By studying wall crossings in Hassett’s moduli spaces of weighted pointed curves, we prove the following.

Theorem 1.2 (Theorem 3.1). Let $g \ge1$ and fix a motivic invariant $\mathbf{H}$ satisfying the blowup formula (2.2). Let $\overline{\mathcal{M}}_{g,r|s}=\overline{\mathcal{M}}_{g,(1^r,(0^+)^s)}$ be the moduli space of stable curves of genus $g$ with $r$ points of weight $1$ and $s$ points of weight $0^+$. Let $\mathbf{H}_{r|s}:=\mathbf{H}_{\overline{\mathcal{M}}_{g,r|s}}$. Then for $r,s\ge0$ with $r+s>0$ and $n>0$, the following hold:

All $\mathbf{H}_{r|s}$ are recursively determined by (1.1) if we know $\mathbf{H}_{0|n}=\mathbf{H}_{\overline{\mathcal{M}}_{g,(0^+)^n}}$ for all $n$.

Next we compute the cohomology of $\overline{\mathcal{M}}_{1,(0^+)^n}$ piece by piece. Let

be the morphism that forgets all the marked points except the first. The inverse image $\mathcal{I}_n=\pi^{-1}(\mathcal{M}_{1,1})$ of $\mathcal{M}_{1,1}$ is the $(n-1)$-fold fiber product of the universal curve

and we have

by the relative Künneth formula, where $V=R^1q_*\mathbb{Q}$. Note that $H_c^*(\mathcal{M}_{1,1},V^{\otimes k})=0$ for $k$ odd by the action of the generic stabilizer $-I\in\operatorname{SL}_2(\mathbb{Z})$. By expanding the right hand side of (3) and applying the Clebsch-Gordan rule, the computation of $H_c^*(\mathcal{I}_n)$ is reduced to that of $H_c^*(\mathcal{M}_{1,1},\operatorname{Sym}^{2j}V)$ which can be obtained by the Eichler-Shimura isomorphism (cf. Lemma 4.2). Combining these, we find that the Hodge-Deligne polynomial of the interior $\mathcal{I}_n$ is (cf. Proposition 4.1)

where

and

The complement $\mathcal{B}_n=\pi^{-1}(\infty)=\overline{\mathcal{M}}_{1,n}-\mathcal{I}_n$ consists of pointed rational nodal curves. Let $\mathcal{B}_{n,j}\subset\overline{\mathcal{M}}_{1,n}$ be the locus of pointed rational curves whose dual graphs are necklaces with $j$ vertices. It is obvious that there are $\gamma_{n,j}$ dual graphs with $j$ vertices, where

denotes the Stirling number of the second kind. For $j\ge2$, $\mathcal{B}_{n,j}\cong(\mathbb{C}^*)^{n-j}$ and hence $H_c^*(\mathcal{B}_{n,j})$ has Hodge-Deligne polynomial $\gamma_{n,j}(uv-1)^{n-j}$. For $j=1,2$, we have to consider an additional action of $S_2$ which acts on $\mathbb{C}^*$ by $z \mapsto z^{-1}$. Combining all these, we find that the Hodge-Deligne polynomial of the boundary $B_n$ is

By (1.4), (1.5) and (1.1), we have the following.

Theorem 1.3 (Theorem 4.4). The Hodge-Deligne polynomial $\mathbf{H}_{0\mid n}$ of $\overline{\mathcal{M}}_{1,(0+)n}=\overline{\mathcal{M}}_{1,0\mid n}$ is

The Hodge-Deligne polynomial $\mathbf{H}_{n\mid0}$ of $\overline{\mathcal{M}}_{1,n}=\overline{\mathcal{M}}_{1,n\mid0}$ can be computed by the recursive formula (1.2) together with (1.1) and (1.6).

Letting $u=v=-t$, we get the Poincaré polynomial of $\overline{\mathcal{M}}_{1,n}$. Note that only the last line in (1.6) has odd degree terms.

It is straightforward to implement the above theorem into a computer program and we can compute the Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal{M}}_{1,n}$ for large $n$ within a relatively short period of time.

Motivated by recent discoveries on the cohomology of $\overline{\mathcal{M}}_{0,n}$ [1, 2, 4, 6, 7, 8], we raise the following.

Question 1.4. (1) Can we refine the computations in this paper to prove a new recursive formula for the $S_n$-equivariant Hodge-Deligne polynomial of $\overline{\mathcal{M}}_{1,n}$?

(2) Is the even degree Poincaré polynomial $P^{\mathrm{even}}_{\overline{\mathcal{M}}_{1,n}}(x)=\sum_{i=0}^{n}x^i\dim H^{2i}(\overline{\mathcal{M}}_{1,n})$ real-rooted?

(3) Do the coefficients of $P^{\mathrm{even}}_{\overline{\mathcal{M}}_{1,n}}(x)$ form a log-concave or $k$-ultra log-concave sequence for some $k$?

(4) Does the probability distribution of the coefficients of $P^{\mathrm{even}}_{\overline{\mathcal{M}}_{1,n}}(x)/P^{\mathrm{even}}_{\overline{\mathcal{M}}_{1,n}}(1)$ converge to a Gaussian distribution?

(5) Can we find effective estimates of the Betti numbers of $\overline{\mathcal{M}}_{1,n}$ for large $n$?

(6) Can we also use the method in this paper to calculate the Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal{M}}_{g,n}$ for $g\ge2$? Note that Theorem 3.1 holds for all $g\ge1$ and hence it suffices to calculate the Hodge-Deligne polynomial of $\overline{\mathcal{M}}_{g,0\mid n}$.

In subsequent papers, we will continue our investigation to seek answers to the above questions.

Convention. All schemes and stacks in this paper are defined over the complex number field $\mathbb{C}$ and all cohomology groups have rational coefficients.

Acknowledgement. It is my pleasure to thank Jinwon Choi and Han-Bom Moon for useful discussions.

Moduli spaces of weighted pointed curves and Hodge-Deligne polynomials

In this section, we collect necessary facts about Hassett moduli spaces [12] and Hodge-Deligne polynomials that will be used throughout this paper.

Definition 2.1. [12] Let $g,n \ge0$. For $\mathcal{A}=(a_1,\cdots,a_n)\in(0,1]^n$ with $2g-2+\sum_i a_i>0$, an integral curve $C$ with marked points $p_1,\cdots,p_n\in C$ is called $\mathcal{A}$-stable if

(1) $C$ has at worst nodal singularities and $p_i$ are all smooth points;

(2) if $p_{i_1}=p_{i_2}=\cdots=p_{i_k}$, then $a_{i_1}+\cdots+a_{i_k}\le1$;

(3) $\omega_C(a_1p_1+\cdots+a_np_n)$ is ample.

Let $\overline{\mathcal{M}}_{g,\mathcal{A}}$ denote the moduli stack of $\mathcal{A}$-stable $n$-pointed curves of genus $g$. In particular, when $a_i>1/2$ for all $i$, $\overline{\mathcal{M}}_{g,\mathcal{A}}$ coincides with the Knudsen-Deligne-Mumford moduli stack $\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$ marked points [9, 15].

Let us summarize useful facts about $\overline{\mathcal{M}}_{g,\mathcal{A}}$ from [12].

Theorem 2.2. Let $\mathcal{A}=(a_1,\cdots,a_n)\in(0,1]^n$ with $2g-2+\sum_{i=1}^n a_i>0$.

(1) $\overline{\mathcal{M}}_{g,\mathcal{A}}$ is a smooth connected proper Deligne-Mumford stack of dimension $3g-3+n$ with projective coarse moduli space.

(2) If $\mathcal{A}=(a_1,\cdots,a_n)$, $\mathcal{A}'=(a'_1,\cdots,a'_n)$ and $a_i\ge a'_i$ for all $i$, then we have a natural morphism

which is obtained by contracting rational components in an $\mathcal{A}$-stable curve which are not $\mathcal{A}'$-stable.

(3) For $2<r\le n$, if $a'_{i_1}+\cdots+a'_{i_r}\le1$ and $a_{i_1}+\cdots+a_{i_r}>1$ but any proper subset of $\{a_{i_1},\cdots,a_{i_r}\}$ has sum at most one while $a_j=a'_j$ for $j\ne i_1,\cdots,i_r$, then the morphism $\rho_{\mathcal{A},\mathcal{A}'}$ is the smooth blowup along the locus $\overline{\mathcal{M}}_{g,\overline{\mathcal{A}}'}$ in $\overline{\mathcal{M}}_{g,\mathcal{A}'}$ where $p_{i_1}=\cdots=p_{i_r}$ and $\overline{\mathcal{A}}'$ is obtained from $\mathcal{A}'$ by replacing $a'_{i_1},\cdots,a'_{i_r}$ by their sum.

(4) If $r\le2$ for any $I\subset[n]$ such that $a_{i_1}+\cdots+a_{i_r}>1$ and $a'_{i_1}+\cdots+a'_{i_r}\le1$, then $\rho_{\mathcal{A},\mathcal{A}'}$ is an isomorphism.

Proof. (1) is [12] and (2) is [12] while (3) is [12] and (4) is [12]. □

Let $\mathbf{H}$ be a motivic invariant with values in an integral domain $R$ which assigns to each variety $X$ over $\mathbb{C}$ an element $\mathbf{H}_X\in R$ such that

(1) $\mathbf{H}_X=\mathbf{H}_{X-Z}+\mathbf{H}_Z$ for any closed $Z\subset X$ and

(2) $\mathbf{H}_{X\times Y}=\mathbf{H}_X\cdot\mathbf{H}_Y$.

If $X\to Y$ is a Zariski locally trivial morphism with $Y$ connected, then $\mathbf{H}_X=\mathbf{H}_Y\cdot\mathbf{H}_F$ where $F$ denotes a fiber.

In this paper, we will only consider motivic invariants satisfying the following blowup formula: If $X\to Y$ is the smooth blowup along $Z$ of codimension $c$, then

Our main example is the Hodge-Deligne polynomial

where $\operatorname{gr}_F^p$ and $\operatorname{gr}_{p+q}^W$ denote the graded parts with respect to the Hodge filtration and weight filtration respectively. By letting $u=v=-t$, we obtain the virtual Poincaré polynomial

of $X$ which is also a motivic invariant.

Reduction to minimally weighted curves

In this section, we prove an explicit formula (Theorem 3.1) that calculates the motivic invariants of $\overline{\mathcal{M}}_{g,n}=\overline{\mathcal{M}}_{g,1^n}$ from those of $\overline{\mathcal{M}}_{g,(0+)^n}$ where $(0+)^n\in(0,1]^n$ denotes an element sufficiently close to zero, for $g\geq1$.

Fix $g\geq1$ and $n\geq1$. Then $\overline{\mathcal{M}}_{g,A}$ is a nonempty proper smooth Deligne-Mumford stack for any $A\in(0,1]^n$. For $0\leq r<n$ and $0\leq s\leq n-r$, let

where 1 is repeated $r-1$ times and $\frac{1}{n-r}$ is repeated $n-r$ times. For instance, $A_{0,s}^{(n)}=\left(\frac{1}{n},\ldots,\frac{1}{n}\right)$ for any $s$. Let

In particular, $Y_{0,n}^{(n)}=\overline{\mathcal{M}}_{g,(0+)^n}$ and

Moreover, by Theorem 2.2 (4),

is the moduli space of stable curves of genus $g$ with $r$ heavy points and $n-r$ light points [13].

Note that if $r>r'$ or $r=r'$ and $s>s'$, then $A_{r,s}^{(n)}-A_{r',s'}^{(n)}\geq0$ and hence we have the natural morphism

by Theorem 2.2 (2).

Fix a motivic invariant $\mathbf{H}$ and let

be the motivic invariant of $\overline{\mathcal{M}}_{g,r|s}$. Then we have $\mathbf{H}_{0|n}=\mathbf{H}_{\overline{\mathcal{M}}_{g,(0+)^n}}$ and $\mathbf{H}_{n|0}=\mathbf{H}_{\overline{\mathcal{M}}_{g,n}}$ by (3.4).

Now our goal is to reach $Y_{n-2,2}^{(n)}=\overline{\mathcal{M}}_{g,n}$ starting from $Y_{0,n}^{(n)}=\overline{\mathcal{M}}_{g,(0+)^n}$ by smooth blowups so that we can reduce the computation of $\mathbf{H}_{n|0}$ to $\mathbf{H}_{0|n}$. Let us first reach $Y_{1,n-1}^{(n)}$. The natural map

is the smooth blowup along the locus $\overline{\mathcal{M}}_{g,1|0}=\overline{\mathcal{M}}_{g,1}$ of $p_1=p_2=\cdots=p_n$ by Theorem 2.2 (3). Likewise, for $1\leq k\leq n-3$, the natural map

is the smooth blowup along the locus of $p_1=p_{i_1}=\cdots=p_{i_{n-k-1}}$ for $\{i_1,\ldots,i_{n-k-1}\}\subset\{2,\ldots,n\}$ which consists of $\binom{n-1}{k}$ copies of $\overline{\mathcal{M}}_{g,1|k}$. Note that $Y_{1,n-2}^{(n)}=Y_{1,n-1}^{(n)}$ by (3.4). By applying the blowup formula (2.2) to these morphisms, we find that

Next, let us reach $Y_{2,n-2}^{(n)}$ from $Y_{1,n-1}^{(n)}$. The natural map

is the smooth blowup along the locus $\overline{\mathcal{M}}_{g,2|0}=\overline{\mathcal{M}}_{g,2}$ of $p_2=p_3=\cdots=p_n$ by Theorem 2.2 (3). For $1\leq k\leq n-4$, the natural map

is the smooth blowup along the locus of $p_2=p_{i_1}=\cdots=p_{i_{n-k-2}}$ for $\{i_1,\ldots,i_{n-k-2}\}\subset\{3,\ldots,n\}$ which consists of $\binom{n-2}{k}$ copies of $\overline{\mathcal{M}}_{g,2|k}$. We have $Y_{2,n-3}^{(n)}=Y_{1,n-2}^{(n)}$ by (12) and the blowup formula (2.2) gives us

Repeating this way, we obtain the following.

Theorem 3.1. For $r,s\geq0$ with $r+s>0$ and $n>0$, the following hold:

All $\mathbf{H}_{r|s}$ are recursively determined by (16) if we know $\mathbf{H}_{0|n}=\mathbf{H}_{\overline{\mathcal{M}}_{g,(0+)n}}$ for all $n$.

In particular, the motivic invariants $\mathbf{H}_{\overline{\mathcal{M}}_{g,n}}=\mathbf{H}_{n|0}$ can be recursively calculated by (16) if we know the motivic invariants $\mathbf{H}_{\overline{\mathcal{M}}_{g,(0+)n}}=\mathbf{H}_{0|n}$ for all $n\geq1$. Conversely, if we know $\mathbf{H}_{\overline{\mathcal{M}}_{g,n}}=\mathbf{H}_{n|0}$ for all $n\geq1$, we can recursively calculate $\mathbf{H}_{\overline{\mathcal{M}}_{g,(0+)n}}=\mathbf{H}_{0|n}$ for all positive $n$ by (16).

Moduli spaces of minimally weighted curves

In this section, we let $g=1$ and let $\mathbf{H}$ be the Hodge-Deligne polynomial. We find an explicit formula that calculates the Hodge-Deligne polynomials of the moduli spaces $\overline{\mathcal{M}}_{1,(0+)^n}$ of stable curves of genus 1 with $n$ marked points whose weights are $\leq1/n$. By Theorem 3.1, we can then calculate $\mathbf{H}_{\overline{\mathcal{M}}_{1,n}}$ for all $n>0$.

Note that in $\overline{\mathcal{M}}_{1,(0+)^n}$, all the marked points are allowed to coincide as long as they stay away from the nodes. Let

be the morphism that forgets all the marked points except the first. Using the stratification

where $\infty$ parameterizes the unique singular rational curve, we have the decomposition

into the interior and the boundary.

Let $\mathcal{I}_n=\mathbf{H}_{\mathcal{I}_n}$ and $\mathcal{B}_n=\mathbf{H}_{\mathcal{B}_n}$ be the Hodge-Deligne polynomials of $\mathcal{I}_n$ and $\mathcal{B}_n$ respectively so that

is the Hodge-Deligne polynomial of $\overline{\mathcal{M}}_{1,(0+)^n}$.

The Hodge-Deligne Polynomial $I_n$ of $\mathcal{I}_n$

In this subsection, we prove the following.

Proposition 4.1. $I_n(u,v)=I_n^{\mathrm{even}}(u,v)+I_n^{\mathrm{odd}}(u,v)$ with

where

See Example 5.1 for an explicit computation of $I_n$ for $n\le10$.

Let $q:\mathcal{C}\to\mathcal{M}_{1,1}$ denote the universal curve. By our choice of stability condition, all the marked points are allowed to coincide and hence $\mathcal{I}_n$ is nothing but the $(n-1)$-fold fiber product

which is a smooth projective fibration with fiber $E^{n-1}$ for an elliptic curve $E\in\mathcal{M}_{1,1}$.

Let $V=R^1q_*\mathbb{Q}$. Then $V$ is a local system of rank 2 over $\mathcal{M}_{1,1}$ and

Let $\pi'=\pi|_{\mathcal{I}_n}$. By the relative Künneth theorem, we have

where $C_k=(\mathbb{Q}\oplus\mathbb{Q}[-2])^{\otimes n-1-k}$.

Since $\mathcal{M}_{1,1}\cong SL_2(\mathbb{Z})\backslash\mathbb{H}$ where $\mathbb{H}$ denotes the upper half plane and the generic stabilizer $-I\in SL_2(\mathbb{Z})$ acts on $V^{\otimes k}$ as $(-1)^k$, we have the vanishing

Hence (4.2) gives us

By the Clebsch-Gordan rule and (4.1), we have

Therefore (4.3) gives us

Since $H_c^*(\mathcal{M}_{1,1},\mathbb{Q})\cong\mathbb{Q}[-2]$, Proposition 4.1 follows immediately from the following.

Lemma 4.2. For $k > 0$, we have an isomorphism

where $S_{2k+2}$ denotes the space of cusp forms of weight $2k+2$ with respect to $SL_2(\mathbb{Z})$. The first factor $\mathbb{Q}$ has weight 0 and $S_{2k+2}$ is of Hodge type $(2k+1,0)$.

Proof. Let $j:\mathcal{M}_{1,1} \to\overline{\mathcal{M}}_{1,1}$ denote the inclusion map. Then we have an exact triangle

By the Eichler-Shimura isomorphism [17], we have

where $S_{2k+2}$ has Hodge type $(2k+1,0)$. We thus find that $H_c^*(\mathcal{M}_{1,1},\mathbb{V}_{2k}) = H_c^1(\mathcal{M}_{1,1},\mathbb{V}_{2k})$ and have an exact sequence

As the cusp $\infty$ has the stabilizer group generated by

$H^0(\infty,j_*\mathbb{V}_{2k}|_{\infty})$ is the $\sigma$-invariant part $(\operatorname{Sym}^{2k}\mathbb{Q}^2)^\sigma\cong\mathbb{Q}$. This completes the proof. □

It is well known that $\dim S_{2k+2} = \beta_{2k+2}$ for $k > 0$ by Riemann-Roch, using the notation of (4.1).

The Hodge-Deligne Polynomial $B_n$ of $\mathcal{B}_n$

Let $\gamma_{n,1}=1$, $\gamma_{n,2}=2^{n-1}-1$ and for $k>2$, let

where $\left\{\begin{matrix} n \\ k \end{matrix}\right\}$ denotes the Stirling number of the second kind, i.e. the number of ways to partition $[n]=\{1,2,\ldots,n\}$ into $k$ unordered subsets. In this subsection, we prove the following.

Proposition 4.3. With $\gamma_{n,k}$ in (4.5),

See Example 5.2 for the calculation of $B_n$ for $n \le10$.

Observe that we have a decomposition

where $B_{n,j}$ denotes the locus of curves with $j$ rational components i.e. the dual graph is a necklace with $j$ vertices.

If $(C,p_1,\cdots,p_n)\in B_{n,1}$, $C$ is a rational curve with normalization $\mathbb{P}^1$. Let $0,\infty\in\mathbb{P}^1$ map to the node of $C$. Then we find that

where $-1\in S_2$ acts by $(z_1,\ldots,z_{n-1})\mapsto(z_1^{-1},\ldots,z_{n-1}^{-1})$. Therefore, we have

whose Hodge-Deligne polynomial is

If $(C,p_1,\ldots,p_n)\in\mathcal{B}_{n,2}$, $C=C_1\cup C_2$ with $C_1,C_2\cong\mathbb{P}^1$. The locus $\mathcal{B}_{n,2,k}$ of such pointed curves where one component has $k$ marked points while the other has $n-k$ points is

where $S_2$ interchanges the two nodes. We thus find that

For $j\geq3$, the number of ways to distribute $n$ marked points to $j$ components is $\gamma_{n,j}$ in (22). For each such distribution, we have a connected component of $\mathcal{B}_{n,j}$ isomorphic to $(\mathbb{C}^*)^{n-j}$. Hence we find that

Adding (23), (24) and (25), we obtain Proposition 4.3.

The Hodge-Deligne polynomial $\mathrm{H}_{\overline{\mathcal{M}}_{1,(0+)^n}}$

By Propositions 4.1 and 4.3, we obtain the following.

Theorem 4.4. The Hodge-Deligne polynomial $\mathrm{H}_{\overline{\mathcal{M}}_{1,(0+)^n}}=\mathrm{H}_{0|n}$ of $\overline{\mathcal{M}}_{1,(0+)^n}$ is

where $\alpha_{ij}$, $\beta_k$ and $\gamma_{n,k}$ are constants defined by (18) and (22) respectively.

See Example 5.3 for a computation of $\mathrm{H}_{0|n}$ for $n\leq10$.

By Theorems 4.4 and 3.1, we can now calculate $\mathrm{H}_{\overline{\mathcal{M}}_{1,n}}(u,v)$ recursively for all $n$.

Computations for n10

Letting $x=uv$, by Proposition 4.1, we can evaluate $I_n$ as follows. Example 5.1.

By Proposition 4.3, we can evaluate $B_n$ as follows with $x=uv$.

Example 5.2.

By adding $I_n$ and $B_n$, we find $T_n=\mathrm{H}_{\overline{M}_{1,0|n}}$ as follows.

Example 5.3.

Note that this matches perfectly with tbl:2 in [13] with $x=uv$.

By (3.8) and Theorem 4.4, we can compute $\mathbf{H}_{r|s}=\mathbf{H}_{\overline{\mathcal{M}}_{1,r|s}}$ as follows with $x=uv$.

Example 5.4.

Note that this matches perfectly with tbl:1 in [13] after applying the hook-length formula to extract the Hodge-Deligne polynomials.

By (3.9) and Theorem 4.4, we can compute $\mathbf{H}_{n|0}=\mathbf{H}_{\overline{\mathcal{M}}_{1,n}}$ as follows with $x=uv$.

Example 5.5.

Note that this matches perfectly with [3, eM1bar.txt] and [10, page 491].

By Theorems 3.1 and 4.4, it is straightforward to write up a computer program that computes the Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal{M}}_{1,n}$ for higher $n$, quite efficiently. For $n=50$, we get the following picture for the probability distribution

The red curve in the picture is the graph of a Gaussian function.

In the subsequent paper [5], we will prove that indeed the probability distribution (5.1) is asymptotically Gaussian. Furthermore, we will prove a formula that gives us effective estimates of the Betti numbers for $n$ large and show that the Betti sequence is asymptotically 3-ultra log concave.

References

School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Korea

Email address: [email protected]