The cotangent bundle of K3 surfaces of degree two

K3 surfaces have been studied from many points of view, but the positivity of the cotangent bundle is not well understood. In this paper we explore the surprisingly rich geometry of the projectivised cotangent bundle of a very general polarised K3 surface $S$ of degree two. In particular, we describe the geometry of a surface $D_S \subset \mathbb{P}(Ω_S)$ that plays a similar role to the surface of bitangents for a quartic in $\mathbb{P}^3$.

💡 Research Summary

The paper investigates the geometry of the projectivised cotangent bundle of a very general polarized K3 surface S of degree two, focusing on the positivity properties of its cotangent bundle Ω_S. While Ω_S is known to be stable for any polarization, its overall positivity is poorly understood; in particular, Ω_S is never pseudoeffective. To measure this negativity the authors study the pseudoeffective cone of the projective bundle π: P(Ω_S) → S, with tautological class ζ_S and pull‑back of a Kähler class α_S.

The authors begin by recalling that for a K3 surface with L² ≥ 8, the divisor ζ_S + πL is pseudoeffective, and that this bound is optimal for very general non‑projective K3 surfaces. They then pose the question of relating pseudoeffectivity of ζ_S + λπL (for λ < L²/8) to the projective geometry of (S,L). In the case of a smooth quartic surface (degree 4) the surface of bitangents U ⊂ P(Ω_S) has class 6ζ_S + 8π*L and generates an extremal ray of the pseudoeffective cone. The goal of the present work is to find an analogue for degree‑two K3 surfaces.

A degree‑two polarized K3 surface (S,L) can be realized as a double cover f : S → ℙ² branched over a smooth sextic curve B; the ramification divisor R∈|3L| is the preimage of B. For a line d⊂ℙ² that is simply tangent to B, its preimage C = f⁻¹(d) is an irreducible curve with a single node; the normalization \tilde C is a smooth elliptic curve. The natural surjection n*Ω_S → Ω_{\tilde C} yields a morphism \tilde C → P(Ω_S), called the canonical lifting of C. Collecting all such liftings for singular elliptic curves in |L| defines a surface D_S ⊂ P(Ω_S).

The first main result (Theorem 1.3) shows that the normalization of D_S is a smooth, non‑minimal elliptic surface. Moreover, the numerical class of D_S is computed as
 D_S ≡ 30 ζ_S + 54 πL = 30(ζ_S + 1.8 πL).
Although D_S is highly singular, its normalization carries rich geometric information about S, including a copy of the branch curve B in its non‑normal locus. However, unlike the quartic case, D_S does not generate an extremal ray of the pseudoeffective cone of P(Ω_S).

The second main result (Theorem 1.4) establishes the existence of a prime divisor Z_S ⊂ P(Ω_S) whose class is a multiple of ζ_S + λπL with λ ≤ 1. Moreover, the canonical liftings of the 324 rational curves in |L| are all contained in Z_S. This is surprising because the non‑nef locus of ζ_S + 1.8 πL consists of a single curve R_S; after blowing up R_S (denoted μ_S : Y → P(Ω_S)), the strict transform of D_S becomes an extremal ray in the pseudoeffective cone of Y, yet D_S itself is a big, non‑nef divisor with negative self‑intersection D_S³ < 0. The divisor Z_S, despite being big, has positive intersection with the lifted rational curves, showing that these curves lie in its stable base locus.

A precise lower bound for λ is given in Theorem 1.5: any prime divisor Z_S with class a(ζ_S + λπ*L) must satisfy λ ≥ 39/22 ≈ 1.772. This bound is far above the trivial λ ≤ 1 from Theorem 1.4, highlighting the subtlety of the problem.

The technical heart of the paper is the analysis of a series of birational maps  Y → P(Ω_S) and Y → P(fΩ_{ℙ²}), which relate the well‑understood projective bundle P(fΩ_{ℙ²}) (identified with the universal family of lines in ℙ²) to the more mysterious P(Ω_S). The authors describe the exceptional divisors E_S and E_P arising from blowing up the curves R_S and R_P (the latter being the projectivisation of the relative cotangent sheaf of the ramification divisor). Detailed intersection calculations (Lemma 3.1) yield the numerical invariants needed for later cone computations.

Through a careful study of the universal family of singular members of |L|, the authors prove that its normalization \bar D is a smooth minimal elliptic surface (Theorem 3.13). They then show that the normalization of D_S is obtained from \bar D by blowing up 720 points (Theorem 3.18), which explains the large discrepancy between the class of D_S and the extremal ray generated by its strict transform on Y.

In the final section the authors connect their constructions to the Hilbert square S^{