Surfaces with canonical map of odd degree

Abstract: Let $S$ be a smooth complex minimal surface of general type with $p_g:=h^0(K_S)\ge 4$ whose canonical map is generically finite of odd degree $d&gt;1$ onto a surface $Σ$. We assume that the general canonical curve of $S$ is smooth and that $Σ$ is ruled by lines, and we prove: - $p_g\le d+2$ - $Σ$ is a cone over the rational normal curve of degree $p_g-2$ in ${\mathbb P}<sup>{p_g-1}$ - $p_g=d+2$ can occur only for $d=3,9,11$. As a byproduct, we refine previous results by Beauville and Xiao by proving that if one drops the assumption that $Σ$ is ruled by lines then $d\le 5$ if $p_g\ge 112$. The case $d=3$ being completely classified by the first two named authors, we focus on $d=5$, showing that $p_g\le 5$ and that for $p_g=5$ the surface $S$ has a pencil $|C|$ with $C<sup>2=1$ and $K_SC=5$. These results suggest that the answer to the question whether the surfaces with canonical map of odd degree $d&gt;1$ have bounded invariants could be positive, in sharp contrast with the case of even degree.

• The paper introduces sharp bounds on the geometric genus, proving that for surfaces with odd degree canonical maps, p_g(S) is bounded by d + 2 with explicit extremal cases.
• It provides a complete classification for the maximum genus case, identifying that only d = 3, 9, or 11 achieve p_g(S) = d + 2 under strict canonical curve conditions.
• The study distinguishes odd degree maps from even degree cases by demonstrating that surfaces with d ≥ 13 cannot have both high invariants and odd degree canonical maps.

Canonical Maps of Odd Degree and Boundedness for Surfaces of General Type

Introduction and Problem Context

The classification of complex algebraic surfaces of general type with prescribed behavior of their canonical maps has been a central topic in the theory of algebraic surfaces. The canonical map, associated to the canonical linear system $|K_S|$ of a smooth complex projective surface $S$, provides critical invariants for the birational geometry of $S$. A longstanding problem focuses on the possible degrees of the canonical map when it is generically finite and, in particular, on the nature and boundedness of numerical invariants when this degree is odd and greater than one.

Prior foundational work by Beauville and Xiao established upper bounds for the degree of the canonical map for surfaces with large geometric genus and provided families of surfaces with canonical maps of even degree and unbounded invariants. However, for odd degrees $d > 1$, explicit constructions and classifications are scarce, and it remains open whether there may exist families with unbounded invariants analogous to those for even degree.

This paper rigorously addresses this gap, concentrating on the structure and possibilities for minimal surfaces of general type $S$ with $p_g(S) = h^0(K_S) \geq 4$ and a generically finite canonical map $\varphi: S \to \Sigma \subset \mathbb{P}^{p_g-1}$ of odd degree $d > 1$, under the crucial assumption that the general canonical curve is smooth and the canonical image $\Sigma$ is ruled by lines.

Main Results

The core contributions can be distilled into the following statements:

1. Sharp Bounds for the Geometric Genus:
   • For a minimal surface $S$ as above, $S$0. Moreover, $S$1 is shown to be the cone over the rational normal curve of degree $S$2 in $S$3.
2. Classification for Maximum Geometric Genus:
   • The extremal case $S$4 is only achievable for $S$5, with the canonical system being base point free, $S$6, and $S$7 for the corresponding pencil $S$8 induced by the lines in $S$9.
3. Exclusion of Higher Odd Degree Cases with Large Invariants:
   • The authors show that for $S$0, $S$1 is sharply bounded by $S$2: no surfaces with both large odd degree canonical maps and unbounded invariants can exist under the hypotheses.
4. In-depth Analysis of Degree 5:
   • For $S$3, only $S$4 is possible, with $S$5 forcing $S$6, $S$7.
5. Contrast With Even Degree:
   • These restrictions for odd degree canonical maps are shown to be in stark contrast with the even degree case, where large families with arbitrarily large invariants are known.

The paper also refines earlier estimates by Beauville and Xiao. For example, if the assumption that $S$8 is ruled by lines is dropped, the canonical degree is still bounded by $S$9 for $d > 1$0.

Technical Methodologies

The proofs blend and generalize a suite of sophisticated techniques:

• Properties of Ruled Surfaces and Canonical Images:
  • By analyzing the geometry and degree of surfaces ruled by lines, particularly cones over rational normal curves, the possible structure of canonical images is rigidly circumscribed.
  • Use of the degree formula, the genus formula, and numerical inequalities such as the Bogomolov-Miyaoka-Yau inequality underpin the bounding arguments.
• Analysis of Linear Systems and Canonical Curves:
  • The presence and structure of pencils $d > 1$1 pulled back from the ruling of $d > 1$2 is essential to the classification, including genus and gonality considerations.
  • Arguments about the connectedness (1-connectedness and 2-connectedness) of curves on minimal surfaces play a crucial role.
• Use of Gap Sequences and Theta Characteristics:
  • Fine analysis of numerical semigroups associated with Weierstrass points and theta characteristics on Gorenstein (possibly reducible) curves, including careful exploitation of invariance of parity in families (using results of Harris and subsequent infinitesimal generalizations).
• Exclusion via Detailed Cohomological Calculations:
  • By computing dimensions of $d > 1$3 for high multiples of the base pencil, and using relative duality, various hypothetical cases are ruled out. For $d > 1$4, all numerical possibilities except $d > 1$5 and $d > 1$6 are excluded by showing impossible dimension counts in the relevant linear systems.
• Construction and Elimination of Hypothetical Examples:
  • For edge cases, notably those corresponding to maximal genus bounds, the existence of putative examples is investigated in detail and ultimately disproven using variation of theta characteristic arguments and dimension theory.

Implications and Theoretical Significance

The results furnish strong evidence that surfaces of general type with non-birational canonical maps of odd degree have bounded invariants, which directly contradicts the situation for even degree. This resolves a previously open question for a significant class of surfaces and corrects earlier beliefs based on the analogy with the even degree case.

Practically, the classification of surfaces with canonical image a cone over a rational normal curve and strict numerical bounds on the geometric genus provide a nearly exhaustive description of the geometry in the odd degree case. The techniques developed, especially in blending the theory of theta characteristics and connectedness properties of curves with surface theory, provide a robust toolkit for future investigations in higher-dimensional analogs or for removing the smoothness assumption on the general canonical curve.

Furthermore, the results illuminate the extent to which parity constraints (odd vs. even degree) fundamentally alter possible geometric and arithmetic configurations in the classification theory of algebraic surfaces.

Future Directions

The methods are, at present, reliant on the assumption that the general canonical curve is smooth. Removing this requirement would be essential for unconditional classifications and likely demands new geometric or deformation-theoretic ideas. Additionally, exploring the analogous bounds and structures for higher-dimensional varieties, or for Gorenstein stable surfaces, remains a promising direction.

Conclusion

This work delivers a comprehensive and rigorous analysis of minimal surfaces of general type with canonical maps of odd degree, culminating in sharp genus bounds and a near-complete classification under natural geometric conditions. The strong contrast with the even degree case underscores the nuanced role of parity in algebraic surface theory and sets a prospective foundation for further research on boundedness phenomena in the birational geometry of surfaces (2604.09216).