Monotone means and a Poisson moving-target limit for complete-intersection curves

Definitions and mathematical statement

The source proposes studying complete intersections and examples in which the target space varies. A complete intersection in ${\mathbb{P}}^n$ is a closed subscheme cut out by a regular sequence of homogeneous equations of the expected codimension; a smooth complete-intersection curve cut out by $n-1$ hypersurfaces has dimension one. The source sentence does not specify a base field, distribution on equations, statistic, limit, or meaning of "moving target space", and therefore gives a research direction rather than a unique quantified conjecture.

The supplied research report studies the following narrower point-count setting. Fix a prime power $q$. For each fixed ambient dimension $n$, consider smooth curves over ${\mathbb{F}}_q$ cut out by $n-1$ hypersurfaces in ${\mathbb{P}}^n$, with the defining degrees tending to infinity in the Bucur--Kedlaya limiting model referenced in the report. Write $\mu_{q,n}$ for the limiting mean of $\#C({\mathbb{F}}_q)$, and $L_{q,n}$ for its reported limiting binomial law. The candidate theorem asserts \[ \mu_{q,1}=q+1,\qquad \mu_{q,n+1}<\mu_{q,n}\quad(n\ge1),\qquad \lim_{n\to\infty}\mu_{q,n}=\lambda_q =(q+1)\prod_{j=3}^{\infty}(1-q^{-j}), \] and \[ \lim_{n\to\infty}d_{\mathrm{TV}} \left(L_{q,n},\operatorname{Pois}(\lambda_q)\right)=0. \] Here $\operatorname{Pois}(\lambda)$ has mass $e^{-\lambda}\lambda^m/m!$ at $m\in{\mathbb{Z}}_{\ge0}$, and total variation on nonnegative integers is $\tfrac12\sum_{m\ge0}|\mu(\{m\})-\nu(\{m\})|$. The dimension-one endpoint corresponds to ${\mathbb{P}}^1$ with no cutting equations. These are iterated limits: first the hypersurface degrees tend to infinity for each fixed $n$, and only then does $n\to\infty$. No simultaneous degree/dimension limit is asserted.

The source record does not provide the explicit fixed-$n$ binomial parameters or the probability measure and degree-growth hypotheses of the Bucur--Kedlaya model. Accordingly the notation $\mu_{q,n}$ and $L_{q,n}$ identifies the quantities in the supplied report, without furnishing a complete independent definition of that model. The report also claims a secondary jet-support theorem giving an exact coefficient-rank obstruction to independent local data for moving-target families; the relevant families, jet orders, linear map and rank formula are absent. That secondary claim cannot be upgraded here to a specific theorem or used to fill in the original meaning of "moving target".

Short English statement

Wood asks for complete-intersection and moving-target examples without specifying a precise statistic. The supplied later report studies point counts of smooth complete-intersection curves over a fixed finite field, claiming strictly decreasing limiting means and an iterated Poisson limit as ambient dimension grows. The underlying sampling model and the separate jet-support theorem still require source review.

Sources

• Local UnsolvedMath registry, permanent ID $20000388$, source code catalogue entry; AIM algebraic number theory corpus, aim-algebraic-number-theory-notes.json, notes[39]. Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0.
• AIM workshop Arithmetic statistics over finite fields and function fields, Question 10, attributed to Melanie Matchett Wood: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf. Original remarks and literature fields are empty. The supplied verification identifies this question but provides no further verification details after its introductory colon.
• The research summary refers to Bucur--Kedlaya complete-intersection limiting laws without a full bibliographic citation or model statement; no missing title, date or formula is invented.
• Registry dated August 14, 2026: partially_solved, PARTIAL-PROGRESS; attempt 1 is labelled partial result, rigor proved, novelty confidence low. Its monotonicity, Poisson-limit and jet-support assertions are machine-generated research aids, not peer reviewed, and require independent verification before citation.