The weight of a weighted filtration is given (for large $m$) by a polynomial

Question

Let $I$ be an homogeneous ideal of $k[x_0, \dots, x_n]$. Suppose to give integral weights $\lambda_0, \dots, \lambda_n$ to $x_0, \dots, x_n$. We assign a weight to every homogeneous polynomial of degree $m$: $$W\left(\sum_{|J|=m} c_J x^J \right) := \max_{c_J \neq 0}\left(\sum_{i=0}^n J_i \cdot \lambda_i\right)$$ Let's define $G_w = \operatorname{span}\left\lbrace P\text{ | }W(P) \leq w\right\rbrace$. We have that $G_{w'} \subset G_w$ if $w' \geq w$. In this way we have a finite number of indices such that there is a "jump" in dimension. We have just constructed a filtration: $$K[x_0, \dots, x_n]^m = G_{w_0} \supsetneq G_{w_1} \supsetneq \dots \supsetneq G_{w_r} \supsetneq 0.$$ Consider now the graduate projection (between components of degree $m$) $\pi_m : k[x_0, \dots, x_n]^m \rightarrow (\frac{k[x_0, \dots, x_n]}{I})^m$. The images of the filtation $\pi(G_{w})$ give us a filtration on the codomain. We call it $F(m)_{w}$ (specifying the degree). To state the proposition we need a definition:

Definition (Weight of a filtration): Let $ G_{w_0} \supsetneq G_{w_1} \supsetneq \dots \supsetneq G_{w_r} \supsetneq 0$ be a weighted filtration on a vector space $V$, we define: $$W_G = \sum_{i=0}^r w_i (\dim G_{w_i}-\dim G_{w_{i+1}})$$ Proposition: Given a homogeneous ideal $I$ and weights $\lambda_0, \dots, \lambda_n$ as above, let $W_F(m)$ be the weight relative to $F(m)_w$:

$W_F(m)$ are given for large $m$ by a numerical polynomial in $m$ of degree $\dim(X) + 1$, where $X= \operatorname{Proj}{\frac{k[x_0, \dots, x_n]}{I}}$.

We could find this statement on page 210 of Harris-Morrison's "Moduli of Curves". It quotes Mumford, "Stability of projective varieties" pp. 39–110, 1977. Mumford uses Chow variety and I would like to avoid it.

The statement has the same flavour as the well-known results on the Hilbert function. I'm confident that there exists a proof that avoids Chow variety.

Where the question comes from:

I'm trying to construct $\mathcal{M}_g$ (the moduli space for smooth curves of genus $g$) following the Gieseker construction in "Moduli of Curves". The difference between the Mumford approach and the Gieseker one is how to prove the stability of smooth curves. Mumford uses Chow variety to show the stability, Gieseker instead avoids it using a different criterion. That's the reason why I'd like to avoid Chow variety.

It's possible to avoid this proposition following the original proof in D. Gieseker, D. R. Gokhale "Lectures on moduli of curves" but, as mentioned before, I'd like to follow the more elegant construction on "Moduli of Curves" (chapter 4).

Does anyone know a proof of the proposition that doesn't use Chow varieties?

Answer 1

I'm aware this is an old question, but I'm answering it for the benefit of anyone who comes across this question in the future.

There are not one but two proofs of this result in the paper Uniform $K$-Stability, Duistermaat-Heckman Measures and Singularities of Pairs by S. Boucksom, T Hisamoto and M Jonsson (their Theorem 3.1); their first proof involves asymptotic Riemann-Roch applied to a sequence of locally trivial fibrations $\pi_d : X_d \to \mathbb{P}^d$ with typical fibre $X$, and their second proof uses equivariant Grothendieck-Riemann-Roch for the $\mathbb{G}_m$-action on the pair $(X, \mathcal{O}_X(1))$ arising from the choice of weights $\lambda_i$.