Exact boundary corrections for stack quotients and branch collisions

Definitions and mathematical statement

John Voight asks when counting formulas become simpler after allowing a partial compactification. A partial compactification is an open immersion $U\hookrightarrow Y$ adding specified boundary strata; $Y$ need not itself be proper. A precise counting question must choose the spaces, allowed boundary, finite fields, ordinary versus stack-weighted counts, and a meaning of "nicer." None is fixed in the short original.

The attached unreviewed aid gives two models. For positive integer weights, let ${\mathbb{P}}(w_0,\ldots,w_m)=[({\mathbb{A}}^{m+1}\setminus0)/\mathbb G_m]$. Its groupoid mass over ${\mathbb{F}}_q$ is the sum of $1/|\operatorname{Aut}(x)({\mathbb{F}}_q)|$ over isomorphism classes of rational objects. The aid reports \[ \#{\mathbb{P}}(w_0,\ldots,w_m)({\mathbb{F}}_q)=\frac{q^{m+1}-1}{q-1}. \] It consequently reports, in characteristic greater than three, mass $q+1$ for the compactified elliptic stack, mass $1$ for its nodal boundary and mass $q$ for the smooth open. These are weighted stack counts, not counts of isomorphism classes with weight one.

For a smooth finite-type curve $C/{\mathbb{F}}_q$, define $Z_C(t)=\prod_{x\in|C|}(1-t^{\deg x})^{-1}$. Let $a_{n,r}$ count effective degree-$n$ divisors whose multiplicities at every closed point are at most $r$, where $r\ge1$. The aid reports \[ \sum_{n\ge0}a_{n,r}t^n=\frac{Z_C(t)}{Z_C(t^{r+1})}. \] For $C={\mathbb{A}}^1$ this gives $a_{n,r}=q^n-q^{n-r}$ for $n\ge r+1$. Thus in monic polynomial coefficient space, for $n\ge3$, replacing the squarefree condition by allowing double roots but no triple roots changes the count from $q^n-q^{n-1}$ to $q^n-q^{n-2}$. The added stratum has $q^{n-1}-q^{n-2}$ coefficient points, and the omitted triple-root locus has codimension two. For equations $y^2=f(x)$, a nodal interpretation additionally needs characteristic not two and suitable curve/compactification conventions; these coefficient counts are not moduli masses or a count of individual nodes.

The aid suggests that independent closed-point factors or simple quotient-stack masses can make boundary corrections explicit. This is a sufficient mechanism illustrated by the two models, not a general characterization of all beneficial partial compactifications.

Short English statement

Identify partial compactifications whose added boundary gives a simpler arithmetic count. The aid supplies weighted elliptic-stack and bounded-root-multiplicity examples, while the general notion of a simpler count remains unspecified.

Sources

• John Voight, Question 11, AIM Arithmetic statistics over finite fields and function fields: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf. Canonical location: aim-algebraic-number-theory-notes.json, notes[40]. The original is labeled exact; remarks and literature are empty. The supplied extraction is used without consulting a PDF.
• The attached boundary-correction aid is machine generated, not peer reviewed, with low novelty confidence. It supplies no complete authoritative references for the stack or divisor formulas.
• Local registry: data/unsolvedmath/problems.json, permanent catalogue entry, exact source code catalogue entry.