Degree Bounds for Canonical Covers of Minimal-Degree Surfaces

Definitions and mathematical statement

In the usual characteristic-zero projective-surface setting, let $X$ be a normal projective surface of general type with canonical singularities, and assume that its canonical divisor $K_X$ is ample and base-point-free. Write $p_g(X)=h^0(X,\omega_X)$. The complete canonical linear system defines a morphism \[ \varphi=\varphi_{|K_X|}:X\longrightarrow W\subseteq{\mathbb{P}}^{p_g(X)-1}. \] The source assumes that the image $W$ is a singular surface of minimal degree. Minimal degree means, in its nondegenerate projective embedding, \[ \deg W=\operatorname{codim}_{{\mathbb{P}}^{p_g(X)-1}}W+1=p_g(X)-2. \] The condition singular is essential; the question is not restricted to a smooth minimal-degree image or to one chosen singular cone.

Since $\varphi^*\mathcal O_W(1)=\omega_X$ is ample, a curve cannot be contracted by $\varphi$; thus the canonical map is finite onto the given surface image in this setting. Its degree is the generic degree \[ \deg\varphi=[k(X):k(W)], \] not the projective degree of $W$. The extracted question is the universal bound \[ \forall\,(X,\varphi,W)\text{ satisfying these hypotheses},\qquad \deg\varphi\leq4\ ? \] Equivalently, under the stated finite-map intersection convention, $K_X^2=(\deg\varphi)\deg W=(\deg\varphi)(p_g(X)-2)$, so the same requested bound is $K_X^2\leq4(p_g(X)-2)$. No Galois, cyclic, flatness, regularity, or smoothness assumption on the cover has been added.

The ground field and characteristic are not explicit in the record; the characteristic-zero setting above is identified as the usual interpretation, not an additional verified report hypothesis. The supplied assessment dated 2026-08-21 describes the extraction as a conditional fragment referring to a preceding canonical-cover question, whereas the supplied original and cleaned text explicitly state the degree-bound question with the displayed hypotheses. That discrepancy must be checked for omitted context. The assessment reports the underlying bound as a research question, not a theorem.

Short English statement

Must a finite canonical map from a canonically singular general-type surface with ample base-point-free canonical divisor have degree at most four when its image is a singular surface of minimal degree? The bound concerns the map, and no Galois-cover restriction is supplied.

Sources

• Local UnsolvedMath registry, permanent ID $30000977$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied.
• Classical Algebraic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/26, pp. 1425--1474. https://doi.org/10.4171/owr/2008/26.
• Supplied reference labelled "recent canonical-cover literature": https://link.springer.com/article/10.1007/s13163-023-00462-5.
• The supplied corrected_verified note reports editorial and LaTeX cleanup without added source-dependent content. The open label and assessment checked 2026-08-21 are dated supplied triage; its conditional-fragment description and the scope of the recent reference require independent source review.