Polynomial Bounds for Rational Maps Between Curves

Definitions and mathematical statement

The source asks whether the number $m(X,Y)$ of rational maps from a curve $X$ to a curve $Y$ is polynomially bounded in the genus $g(X)$. It gives no field, smoothness or projectivity hypotheses, genus restriction on $Y$, or convention for which maps are counted. These omissions affect finiteness before they affect a quantitative bound.

Under the usual illustrative convention of smooth projective connected curves over ${\mathbb{C}}$, the genus is $g(X)=\dim_{{\mathbb{C}}}H^0(X,\Omega_X^1)$. A rational map is a morphism on a nonempty Zariski-open subset, with two representatives identified when they agree on a nonempty open subset. Let $\mathcal R(X,Y)$ stand for the collection intended by the source, and write $m(X,Y)=|\mathcal R(X,Y)|$. This notation leaves unresolved whether $\mathcal R$ means all rational maps, only nonconstant maps, or maps modulo a specified equivalence; the draft does not silently substitute one of these choices.

For a specified admissible class $\mathcal A$ of pairs $(X,Y)$, a uniform polynomial-bound interpretation is \[ \exists C>0\ \exists d\in{\mathbb{Z}}_{\geq0}\ \forall (X,Y)\in\mathcal A, \qquad m(X,Y)\leq C(1+g(X))^d. \] If the target is fixed instead, the quantifiers might be $\forall Y\ \exists C_Y,d_Y\ \forall X$, or the degree might be required to be absolute while only the constant depends on $Y$. The source does not distinguish these requests. The term $1+g(X)$ is only a convenient convention for expressing a polynomial growth bound at all nonnegative genera.

If all constant maps between complex curves are counted, the points of $Y({\mathbb{C}})$ already supply infinitely many maps. Merely excluding constants still leaves an essential target-genus issue: unrestricted maps to ${\mathbb{P}}^1$ and elliptic targets need not form finite collections. These are missing-hypothesis warnings, not a replacement conjecture or a claimed resolution of the intended problem. Restrictions such as a hyperbolic target, a degree bound or an automorphism quotient cannot be inferred from the extracted wording.

The literature assessment checked 2026-08-21 says only a lead-in survives and the conjecture itself is missing, whereas the supplied original and clean fields both contain the displayed map-counting question. The verification note reports restoration of notation only. This source mismatch and the supplied open label require review; neither furnishes the missing hypotheses or a current theorem-status conclusion.

Short English statement

The intended question asks for a polynomial bound, in the source genus, on a count of maps to another curve. A meaningful finite count needs the field, curve hypotheses, nonconstant-map convention, target restrictions and uniformity of the bound. These data are absent, and the literature note appears to describe a different or earlier extraction.

Sources

• Local UnsolvedMath registry, permanent ID 30000185, source code catalogue entry, "Polynomial Bounds for Rational Maps Between Curves," including the original and cleaned question, notation-restoration verification and conflicting extraction assessment.
• Komplexe Algebraische Geometrie (2005), Oberwolfach Reports, pp. 407--463, DOI 10.4171/owr/2005/07: https://doi.org/10.4171/owr/2005/07. This is the sole supplied literature reference, labelled "original OWR report."
• Literature triage checked 2026-08-21; registry record updated 2026-09-16. No external source has been visited or independently reviewed here.