A level-cover ramification budget for rational and affine coarse curves in M_g

Definitions and mathematical statement

Let $M_g$ denote the coarse moduli space of smooth projective curves of genus $g$, in the complex $g\geq2$ interpretation used by the aid, and distinguish it from its stack $\mathcal M_g$ and stable compactification. The question asks whether $M_g$ contains a ${\mathbb{P}}^1$, or an ${\mathbb{A}}^1$. "Contains" may mean an embedded subvariety or merely the image of a nonconstant morphism; these are different assertions. The source does not fix $g$, field or embedding convention. It reports algebraic hyperbolicity of the stack while expecting different behavior for its coarse space, without specifying a hyperbolicity definition.

The aid reports that Oort proved existence of a complete rational coarse curve for some genus, and that Bryan--Donagi--Stipsicz constructed nonconstant morphisms \[ {\mathbb{P}}^1\longrightarrow M_{3n^3-n^2+1}\quad(n\geq2), \] starting with genera $21,73,177,351$. Restriction then gives nonconstant ${\mathbb{A}}^1$-maps in those infinitely many genera. It explicitly says these references do not automatically supply embedded smooth ${\mathbb{P}}^1$ or ${\mathbb{A}}^1$ subvarieties, and that it found no theorem covering every fixed genus. That search limitation is not a new universal nonexistence or open-status theorem.

Its obstruction lemma takes a nonconstant morphism $f:B={\mathbb{P}}^1\setminus S\to M_g$, with $S$ finite, and a full-level-$N$ cover, $N\geq3$, admitting the usual fine-moduli interpretation. Let $C^\circ\to B$ be a normalized connected Galois component of its pullback, or the specified Galois cover, and let $b_1,\ldots,b_v\in B$ be its internal branch points with actual ramification indices $m_i$. The aid reports \[ \sum_{i=1}^v\left(1-\frac1{m_i}\right)>2-|S|. \] Equivalently its compactification $C$, punctured above $S$, has positive logarithmic canonical degree, with logarithmic Riemann--Hurwitz normalized by the cover degree. Branching above the removed set $S$ is not included again in the internal sum. In particular the stated internal budget is greater than two for $B={\mathbb{P}}^1$ and greater than one for $B={\mathbb{A}}^1$.

For a local cyclic quotient arc $x=t^a$, $a\geq1$, and coarse contact $x=z^c$, $c\geq1$, the aid reports normalized effective index \[ m=\frac a{\gcd(a,c)}. \] Thus the budget uses actual ramification, not an unadjusted stabilizer order. This cyclic arc formula does not assert every higher-dimensional stack quotient is of that form. The obstruction constrains a map if it exists and does not settle embedded curves or every fixed genus.

Short English statement

Ask whether coarse curve moduli contain projective or affine lines. The attached aid reports nonconstant maps in infinitely many genera, distinguishes those from embedded lines, and gives a level-cover ramification budget using actual contact-adjusted indices.

Sources

• Local UnsolvedMath registry, permanent ID $20000345$, exact public code catalogue entry; canonical location aim-algebraic-geometry-notes.json, notes[344].
• AIM workshop "Compact moduli spaces and birational geometry," Problem 5.3; supplied URL https://aimath.org/WWN/birational/birational.pdf. No proposer or year is supplied; the literature field is empty. The PDF was not consulted here.
• Attached research attempt 1, "A level-cover ramification budget for rational and affine coarse curves in M_g," attributing existence results to Oort and Bryan--Donagi--Stipsicz without full publication citations.