Ordered-root graph filtration for the symmetric quotient of a product of projective lines

Symmetric quotient and monic tower. The product of the linear binary forms vanishing at the $n$ tuple entries gives a point of the projective space of degree-$n$ binary forms. A nonzero complex binary form factors into such forms, uniquely as an unordered multiset, so its fibres are exactly the permutation orbits. On the open set of forms nonvanishing at $a$, normalize the leading coefficient to one. Its coefficient ring is $A=\mathbb{C}[e_1,\dots ,e_n]$ and the preimage has ring $R=\mathbb{C}[x_1,\dots ,x_n]$, where $e_j$ are the elementary symmetric polynomials. These generate all symmetric polynomials: subtract the product of elementary symmetric polynomials with the same largest monomial of a symmetric polynomial in lexicographic order; its exponents are weakly decreasing, and the needed product is $e_1^{a_1-a_2}\cdots e_n^{a_n}$. Subtraction strictly decreases the largest monomial among the finitely many monomials of bounded total degree, so induction terminates. Their algebraic independence follows because every coefficient tuple occurs as a monic polynomial with complex roots: a polynomial relation would vanish on all of ${\mathbb{C}}^n$, hence be zero by induction in the variables. Thus this chart is the affine invariant quotient.

The universal splitting algebra over $A$ is built by adjoining $t_1$ with $P(t_1)=0$, then $t_2$ with $(P(T)\mathrm{/}(T-t_1))(t_2)=0$, and so on. The successive polynomials are monic of degrees $n,n-1,\dots ,1$; polynomial division gives their coefficients in the preceding quotient, because the remainder is zero. This algebra is isomorphic to $R$: comparison of $P(T)={\prod }_j(T-t_j)$ recovers each coefficient $e_j$ as the elementary symmetric polynomial in the $t_j$, and gives inverse presentations. Therefore $R$ is finite free of rank $n!$ over $A$; its basis is $t_1^{a_1}\cdots t_n^{a_n}$ with $0\le a_j\le n-j$. Every fibre is nonempty by factorization, and the rank is positive, so the map is faithfully flat. These quotient charts cover ${\mathbb{P}}^n$: choose $a$ outside the finite root multiset of a given form. Their invariant descriptions agree on overlaps because both are the same product binary form. They prove the global quotient identification and the finite faithfully flat degree. The monic dual-pairing argument in F1 verifies the dual-line hypothesis for this map, including repeated-root fibres.

The elementary Newton filtration. Work over $R=\mathbb{C}[x_1,\dots ,x_n]$. The fibre-product algebra is the splitting algebra $C=R{\otimes }_{A}R$ for $P(T)={\prod }_i(T-x_i)$, with the second ordered roots denoted $y_1,\dots ,y_n$. Set $B=R[t]\mathrm{/}P(t)$ and $q_i(t)={\prod }_{k<i}(t-x_k)$ for $1\le i\le n+1$. The ideals $q_{i}B$ form a descending filtration, with first term $B$ and last term zero. Multiplication by $q_i$ identifies $R[t]\mathrm{/}{\prod }_{k\ge i}(t-x_k)$ with $q_{i}B$: in $R[t]$, the equality $P=q_i{\prod }_{k\ge i}(t-x_k)$ and cancellation of the nonzero monic polynomial $q_i$ prove both its kernel and injectivity. Hence $q_{i}B\mathrm{/}{q}_{i+1}B\cong R[t]\mathrm{/}(t-x_i)=R\mathrm{.}$ Restriction to the root $t=x_i$ sends the indicated generator to $d_i={\prod }_{k<i}(x_i-x_k)$, so its image is exactly the ideal $d_{i}R$. This restriction is injective on the quotient because $R$ is a domain and $d_i\mathrm{\ne }0$. Also $q_{i}B$ is the kernel of restrictions to $t=x_1,\dots ,x_{i-1}$: induction divides a polynomial vanishing at these roots by $t-x_1$, then evaluates at $x_2$ and cancels $x_2-x_1$, and continues. This proves the kernel assertion over the polynomial ring even where individual roots coincide; it does not localize away from diagonals.

An explicit triangular basis, checking the entire refined filtration. For a permutation $w$, put $A_j(w)=\{k<w_j:k\mathrm{\notin }\{w_1,\dots ,w_{j-1}\}\}$ and define $N_w(y;x)={\prod }_j{\prod }_{k\in A_j(w)}(y_j-x_k)\in C$. The integers $a_j(w)=\mathrm{\mid }{A}_j(w)\mathrm{\mid }$ satisfy $0\le a_j(w)\le n-j$. Conversely such integers recover $w$ uniquely: at stage $j$ choose the $(a_j+1)$st unused label in increasing order. Thus $w\mapsto (a_1,\dots ,a_n)$ is a bijection to the exponent vectors of the monic-tower basis of step 2.1. In that basis, $N_w=y_1^{a_1}\cdots y_n^{a_n}$ plus terms of strictly smaller total $y$-degree (all exponents already stay in the allowed ranges). The change-of-basis matrix, ordered by total degree, is triangular with diagonal one. Therefore the $N_w$ form an $R$-basis of $C$, with no cited Schubert-basis theorem. If $v$ is lexicographically earlier than $w$, their first differing index $j$ has $v_j<w_j$ and $v_j$ unused after the identical prefix; hence the factor $y_j-x_{v_j}$ of $N_w$ vanishes on $G_v$. On its own graph $G_w$, its value is exactly $d_w={\prod }_j{\prod }_{k\in A_j(w)}(x_{w_j}-x_k)$. These are nonzero polynomials in the domain $R$. Write any element $c={\sum }_w{b}_w{N}_w$. Vanishing on the first graph forces its first coefficient zero by cancellation of $d_w$; vanishing on the second then forces its second coefficient zero; induction proves that the kernel of the first $r$ graph restrictions is exactly the $R$-span of the remaining $N_w$. Its next quotient therefore restricts isomorphically to $d_{w}R$ on $G_w$, and the final kernel is zero. Since these are kernels of actual algebra restriction maps, they are ideals and hence sub-bimodules; the quotient isomorphism respects both coordinate-ring actions. This proves precisely the asserted scheme-theoretic filtration, retaining repeated-root fibres.

Global gluing without a homogenization assumption. The kernels $K_r$ are defined globally as kernels of the actual graph-restriction maps; consequently their restrictions to all the affine charts just used agree. These charts cover $Z$: both tuples in a point of $Z$ have the same finite root multiset, so some $a$ is avoided by both tuples. On each such chart the quotient image is exactly the product of the equations of the pair diagonals appearing in $D_w$, by step 4.1. Under a change of affine coordinate $u\mapsto (\alpha u+\beta )\mathrm{/}(\gamma u+\delta )$, each difference changes by the unit $\det (\alpha ,\beta ;\gamma ,\delta )\mathrm{/}((\gamma x_i+\delta )(\gamma x_k+\delta ))$ on the common chart. These are exactly the transitions of the ideal of the corresponding Cartier pair diagonal. Their product is therefore the ideal line ${\mathcal{O}}_Y(-D_w)$, rather than an unspecified degree shift. Each pair diagonal on $({\mathbb{P}}^1{)}^n$ has equation $X_{i,0}{X}_{k,1}-X_{i,1}{X}_{k,0}$ and line $\mathcal{O}(1)$ on those two factors, so it is effective Cartier and contributes degree one in each. A label can pair with at most $n-1$ other labels. This proves all global filtration and degree assertions.

Derived functor consequence. Use the first copy of $Y$ for $y$, the second for $x$. The finite-flat module identity identifies $f^{\ast }{f}_{\ast }E=R{p}_{x\ast }(p_y^{\ast }E{\otimes }^{\mathbf{L}}{\mathcal{O}}_Z)\mathrm{.}$ The filtration triangles of ${\mathcal{O}}_Z$ may be tensor-pushed by this exact derived functor. Each graph projection to either copy is an isomorphism, so the graph factor is the pullback of $E$ under $x\mapsto (x_{w_1},\dots ,x_{w_n})$, tensored with ${\mathcal{O}}_Y(-D_w)$; there is no hidden Tor term on a graph, as a finite locally free resolution computes the derived tensor there. $p_x$ is finite on all these supports, so its pushforward is acyclic by F1. For $E\in P(\theta )$ every graph pullback has phase $\theta$ by the assumed permutation invariance, and every negative twist lies in $P(\le \theta )$ by hypothesis. Finite extensions retain that bound, proving $f^{\ast }{f}_{\ast }P(\theta )\subset P(\le \theta )$. Apply F1 to obtain the entire induced stability condition and actual image-lattice support property.