Hilbert series and eventual Hilbert value of a two-form plane complete intersection

Fix $n$. Because $F$ is a nonzerodivisor on $S$ of degree $d$ by [L2], multiplication by $F$ maps $S_{n-d}$ isomorphically onto $F\cdot S_{n-d}=(F{)}_n$, so there is an exact sequence of $k$-vector spaces $0\to S_{n-d}\stackrel{\cdot F}{\to }{S}_n\to (S\mathrm{/}(F){)}_n\to 0.$ Here $S_{n-d}:=0$ when $n<d$. Likewise $G$ is a nonzerodivisor on $S\mathrm{/}(F)$ and has degree $e$, so $0\to (S\mathrm{/}(F){)}_{n-e}\stackrel{\cdot G}{\to }(S\mathrm{/}(F){)}_n\to (S\mathrm{/}(F,G){)}_n\to 0$ is exact, with $(S\mathrm{/}(F){)}_{n-e}:=0$ when $n<e$. All terms are finite-dimensional over $k=S_0$, and ${\mathrm{\ell }}_k(V)={\dim }_{k}V$ for a finite-dimensional $k$-vector space, since a basis $v_1,\dots ,v_m$ gives the composition series $0⊊⟨v_1⟩⊊\cdots ⊊V$ with one-dimensional, hence simple, factors by [L6].

Taking dimensions over $k$ in the two exact sequences of 1.1 and using ${\mathrm{\ell }}_k={\dim }_k$ by [L4] on each piece, we get for every $n$ ${\dim }_k(S\mathrm{/}(F,G){)}_n={\dim }_k{S}_n-{\dim }_k{S}_{n-d}-{\dim }_k{S}_{n-e}+{\dim }_k{S}_{n-d-e},$ with ${\dim }_k{S}_m:=0$ for $m<0$. Moreover ${\dim }_k{S}_m$ is the number of triples $(a,b,c)\in {\mathbb{N}}^3$ with $a+b+c=m$, since those triples index the monomial basis of $S_m$ by [L3].

Work in $\mathbb{Z}\mathrm{⟦}t\mathrm{⟧}$, so the coefficients retain the integer dimensions even when $k$ has positive characteristic. Write $G(t):={\sum }_{n\ge 0}{\dim }_k{S}_n{t}^n$ for the Hilbert series of $S$. By the Cauchy product rule of [L7], the cube of ${\sum }_{n\ge 0}{t}^n$ is ${\sum }_n{t}^n$ convolved three times, whose coefficient at $t^n$ is exactly the number of triples $(a,b,c)\in {\mathbb{N}}^3$ with $a+b+c=n$, that is, ${\dim }_k{S}_n$ by 2.1. Hence $G(t)=({\sum }_{n\ge 0}{t}^n{)}^3$; and since $1-t$ times ${\sum }_{n\ge 0}{t}^n$ has constant coefficient one and all other coefficients zero, ${\sum }_{n\ge 0}{t}^n$ is the inverse of $1-t$ in $\mathbb{Z}\mathrm{⟦}t\mathrm{⟧}$ and $G(t)=(1-t{)}^{-3}$. Multiplying the dimension identity of 2.1 by $t^n$ and summing over $n\ge 0$, the shifts by $d$ and $e$ contribute $t^d$ and $t^e$ by the twist rule of [L4], so ${\mathrm{HS}}_{S\mathrm{/}(F,G)}(t)=(1-t^d)(1-t^e)G(t)=\frac{(1-t^d)(1-t^e)}{(1-t{)}^3}\mathrm{.}$

Since $(1-t^d)=(1-t)(1+t+\cdots +t^{d-1})$ and likewise for $e$, the series of 3.1 equals $P(t)\mathrm{/}(1-t)$ where $P(t)$ is the polynomial $(1+t+\cdots +t^{d-1})(1+t+\cdots +t^{e-1})={\sum }_{m=0}^{d+e-2}{p}_m{t}^m$ with $p_m\ge 0$; here $p_m$ counts the pairs $(i,j)$ with $i\le d-1$, $j\le e-1$ and $i+j=m$, so ${\sum }_m{p}_m=de$. By the Cauchy product rule of [L7] and the inverse ${\sum }_n{t}^n=(1-t{)}^{-1}$ from 3.1, the coefficient of $t^n$ in $P(t)\mathrm{/}(1-t)$ is ${\sum }_{m\le n}{p}_m$, which equals ${\sum }_m{p}_m=de$ for every $n\ge d+e-2$. Hence $H_{S\mathrm{/}(F,G)}(n)=de$ in all those degrees.

By [L1], the hypothesis of the statement holds in particular for every pair of nonzero homogeneous plane forms of positive degrees $d,e$ with no common nonconstant factor, so the computed series and the eventual value $de$ apply to those pairs. Steps 3.1 and 4.1 prove both displayed identities and the eventual constancy; no Artinianity or finite dimensionality of $S\mathrm{/}(F,G)$ was used anywhere.