Positivity of tangent bundle of weak Del Pezzo surfaces of degree $4$

Abstract: In this paper, we prove that for any weak Del Pezzo surface $S$ of degree at least $4$, the tangent bundle $T_S$ is almost nef. For the proof, we use total dual VMRTs induced by conic bundle structures.

• The paper proves that the tangent bundle of any weak Del Pezzo surface of degree 4 is almost nef, resolving a boundary case in positivity classification.
• It employs a detailed geometric construction using conic bundle structures, effective divisor constructions, and VMRT analysis to assess positivity.
• The work provides a complete characterization of the conditions for almost nefness, setting grounds for further studies on higher-dimensional positivity.

Positivity of the Tangent Bundle for Weak Del Pezzo Surfaces of Degree 4

Introduction and Problem Formulation

The paper "Positivity of tangent bundle of weak Del Pezzo surfaces of degree $4$" (2604.09300) presents a definitive result on the almost nefness of the tangent bundle $T_S$ for any weak Del Pezzo surface $S$ of degree at least 4, with an explicit focus on the borderline case $d=4$. The study is set within the context of the classification of positivity properties of tangent bundles for projective varieties, with a focus on the implications for birational and differential-geometric structure.

Recall that Mori's theorem characterizes smooth projective varieties with ample tangent bundles as isomorphic to projective spaces. Weaker positivity notions—such as nefness, almost nefness, and pseudo-effectivity—have since been systematically investigated, especially for surfaces. Existing work (e.g., Campana-Peternell, Demailly-Peternell-Schneider) classifies surfaces with nef $T_S$, while recent studies have elucidated the minimal degree at which pseudo-effectivity or almost nefness arises for (weak) Del Pezzo surfaces. In particular, for a Del Pezzo surface of degree $d$, one has $T_S$ almost nef if and only if $d\geq 4$, with $d=4$ previously standing as a subtle, structurally intricate boundary case.

Main Results

The central result of this work is:

Theorem: For any weak Del Pezzo surface $S$ of degree 4, the tangent bundle $T_S$0 is almost nef.

The proof strategy provides a detailed geometric construction, leveraging conic bundle structures and intricate properties of the projectivized tangent bundle $T_S$1. The notion of almost nefness employed is that $T_S$2 is nef on every curve outside a countable union of proper closed subsets.

The result extends immediately to degrees $T_S$3 and establishes a dichotomy: weak Del Pezzo surfaces of degree $T_S$4 do not exhibit almost nef $T_S$5, while those of degree $T_S$6 do.

Technical Approach

Birational and Positivity Techniques

The argument invokes the geometry of conic bundles on weak Del Pezzo surfaces. The core geometric input is the construction of effective divisors on $T_S$7 of the form $T_S$8, where $T_S$9 is the tautological bundle and $S$0 is the projection. The divisors $S$1 arise as total dual varieties of minimal rational tangents (VMRTs) associated with suitable conic bundle structures.

For strict (i.e., smooth) Del Pezzo surfaces, the positivity of $S$2 is controlled by a decomposition of $S$3 into total dual VMRTs. Weak Del Pezzo surfaces of degree $S$4 lack such a uniform decomposition, especially when blowing up points not in "good position" (i.e., configurations where no three out of certain four points are colinear in a generalized sense).

The analysis proceeds by classifying all conic bundle structures (degree 1 or 2) via a combinatorial and geometric study of the blowup morphisms $S$5, exceptional sets, and the corresponding Picard and Néron-Severi lattices. The existence of conic bundles of degree 2 is shown to be equivalent to the presence of four blown-up points in good position.

Sufficient Condition for Almost Nefness

A key technical criterion established is:

• If there exists an effective divisor $S$6 on $S$7 such that $S$8 is pseudo-effective, then $S$9 is almost nef.

The proof examines the possible configurations of singular fibers in such conic bundles (with controlled multiplicity), and employs intersection-theoretic bounds to check pseudo-effectivity of restrictions.

Case Analyses: Good Position and Non-Good Position

If four blown-up points are in good position (in the sense of the generalized colinearity notion), the construction yields conic bundles $d=4$0 and $d=4$1 (of degrees 1 and 2), inducing divisors $d=4$2 and $d=4$3 with the required positivity. A detailed sum formula shows that the sum $d=4$4 is effective, ensuring almost nefness by the aforementioned criterion.

If no such four points exist, the geometry ensures the existence of a conic bundle structure $d=4$5 (of degree 1) for which the relative tangent bundle $d=4$6 is effective, again meeting the required criterion.

Novel Claims and Numerical Bounds

The paper's significant claim is the universal almost nefness of $d=4$7 for every weak Del Pezzo surface $d=4$8 of degree 4, including all possible combinatorial configurations of exceptional loci (chains of $d=4$9- and $T_S$0-curves due to the weak structure). This settles open cases in the minimal model program for surfaces regarding near-positivity of tangent bundles.

Key numerical inequalities are proved, for example:

• The sum over the singular fibers $T_S$1 of $T_S$2 for a conic bundle structure is always $T_S$3, where $T_S$4 denotes the maximal multiplicity in the fiber.
• The necessary combinatorial distinctions between cases depending on the structure (i.e., number and arrangement of singular fibers and exceptional curves) are fully explored, removing ambiguities about the boundary case $T_S$5.

Implications and Future Directions

This result advances the program of classifying varieties (and, in particular, surfaces) according to positivity properties of their tangent bundles, enriching the inventory of examples and non-examples at the precise boundary for Del Pezzo surfaces. The explicit constructions and combinatorial techniques suggest further investigations into analogous boundary cases for higher-dimensional Fano or weak Fano varieties, as well as links with rational connectedness and VMRT theory.

The method underscores the nuanced role of almost nefness as an intermediate positivity property—strictly weaker than nefness, but strong enough to infer substantial geometric and birational consequences. The techniques could inform further explorations into the relationship between almost nef vector bundles and rational or weakly Fano geometry, as well as questions regarding the structure of moduli spaces and automorphism groups for rational surfaces.

The approach also raises questions on the connections between the existence of Lagrangian fibrations in the cotangent/canonical bundle setting and almost nefness of the tangent bundle, especially in view of the cited work on cotangent bundles of del Pezzo surfaces of degree 4.

Conclusion

The paper establishes that the tangent bundle $T_S$6 of any weak Del Pezzo surface $T_S$7 of degree at least 4 is almost nef, providing a complete answer for this class of surfaces. The analysis is characterized by a comprehensive case-by-case breakdown, involving detailed birational and divisor-theoretic computations on projective bundles over $T_S$8. These results close a significant gap in the classification of almost nef tangent bundles on rational surfaces, and lay groundwork for further exploration of positivity in more general classes of higher-dimensional varieties.