For a rank-two vector bundle EE on a complex Fano manifold XX of Picard number one, the nef cone of the projectivization P(E)\mathbb{P}(E) is a two-dimensional closed cone whose non-fiber boundary ray has slope τ(E)\tau(E), while τ0(E)\tau_0(E) is the supremum of the slopes computed on rational curves of XX from their splitting types. Muñoz–Occhetta–Solá Conde, who introduced these invariants, knew of no example with τ≠τ0\tau\ne\tau_0 and asked whether equality always holds, i.e. whether the positivity of the bundle, as encoded by the nef cone of its projectivization, is always detectable on rational curves. Equality is known for split bundles, for Fano bundles (τ<iX\tau<i_X), for the Horrocks–Mumford bundle on P4\mathbb{P}^4, and for the Schwarzenberger bundle at very general sets of 99 points of P2\mathbb{P}^2; on P2\mathbb{P}^2 the question is intertwined with Nagata's conjecture on Seshadri constants. The question remains open.

Background

A Fano manifold is a smooth complex projective variety with ample anticanonical class. Throughout, XX denotes a Fano manifold of Picard number one (its Néron–Severi group has rank one), with Pic(X)=Z⋅[HX]\mathrm{Pic}(X)=\mathbb{Z}\cdot[H_X], and iXi_X its index, the integer with −KX∼iXHX-K_X\sim i_X H_X. Let EE be a normalized rank-two vector bundle on XX, i.e. det⁡E≃OX(c1HX)\det E\simeq\mathcal{O}_X(c_1H_X) with c1∈{0,−1}c_1\in\{0,-1\}. On the Grothendieck projectivization P(E)=ProjX(⨁kSkE)\mathbb{P}(E)=\mathrm{Proj}_X(\bigoplus_k S^kE) (parametrizing one-dimensional quotients, so that the tautological line bundle O(1)\mathcal{O}(1) is a quotient of π∗E\pi^*E), write HH for the pullback of HXH_X, LL for the tautological divisor, and −Krel=2L−c1H-K_{\mathrm{rel}}=2L-c_1H for the relative anticanonical divisor, so that −KP(E)=−Krel+iXH-K_{\mathbb{P}(E)}=-K_{\mathrm{rel}}+i_XH. The Néron–Severi space N1(P(E))RN^1(\mathbb{P}(E))_\mathbb{R} is two-dimensional, spanned by the classes of HH and LL; its nef cone (the closed cone of divisor classes intersecting every curve nonnegatively) is therefore a planar closed cone bounded by the fiber class and one further boundary ray. Muñoz–Occhetta–Solá Conde defined τ(E)\tau(E) as the slope of that cone: the unique real number tt such that −Krel+tH-K_{\mathrm{rel}}+tH is nef but not ample, and dually ρ(E)\rho(E) as the slope of the pseudoeffective cone (the closure of the cone of effective divisors). The source paper proves that τ≥0\tau\ge 0, with equality exactly for the trivial bundle OX⊕2\mathcal{O}_X^{\oplus 2}. The invariant is accessible on curves: if C⊂XC\subset X is a curve such that E∣CE|_C is not semistable (semistable means that no line subbundle of the rank-two bundle E∣CE|_C has degree larger than half of deg⁡E∣C\deg E|_C), the minimal sections of P(E)\mathbb{P}(E) over CC — the sections of the projection P(E)→X\mathbb{P}(E)\to X over CC on which the tautological divisor LL has minimal degree, which exist precisely in the non-semistable case — have a computable slope τ(C)\tau(C), and, for nontrivial EE, τ\tau equals the supremum of τ(C)\tau(C) over all such curves. For a rational curve ℓ\ell with splitting type E∣ℓ~≃O(a)⊕O(b)E|_{\tilde\ell}\simeq\mathcal{O}(a)\oplus\mathcal{O}(b) on its normalization ℓ~≃P1\tilde\ell\simeq\mathbb{P}^1 (the degrees of the line-bundle summands), one gets τ(ℓ)=∣b−a∣/(HX⋅ℓ)\tau(\ell)=|b-a|/(H_X\cdot\ell). Since Mori theory (the study of projective varieties through their rational curves) shows that Fano manifolds are covered by rational curves, a central question of the theory — explicitly posed by the source paper — is to what extent the geometry of EE is determined by its behavior on rational curves; this motivated the definition τ0(E):=sup⁡{τ(C)∣C⊂X rational curve}\tau_0(E):=\sup\{\tau(C)\mid C\subset X\text{ rational curve}\}. Several cases are known: for split bundles τ=∣b−a∣\tau=|b-a| is computed by minimal sections over rational curves; for a Fano bundle, meaning a bundle with P(E)\mathbb{P}(E) Fano (equivalently −KP(E)-K_{\mathbb{P}(E)} ample, equivalently τ<iX\tau<i_X), Mori theory gives τ∈Q\tau\in\mathbb{Q}, semiampleness of −Krel+τH-K_{\mathrm{rel}}+\tau H, and a rational curve orthogonal to the boundary, whence τ=τ0\tau=\tau_0; the Horrocks–Mumford bundle (a classical indecomposable rank-two bundle on P4\mathbb{P}^4) has τ=τ0=7\tau=\tau_0=7; and the Schwarzenberger bundle associated with kk points P⊂P2P\subset\mathbb{P}^2 (the rank-two bundle whose pullback to the blow-up of P2\mathbb{P}^2 at PP is an extension of O(−E)\mathcal{O}(-E) by O(E)\mathcal{O}(E), EE the exceptional divisor) satisfies 2/τ=ε(OP2(1);P)2/\tau=\varepsilon(\mathcal{O}_{\mathbb{P}^2}(1);P), the Seshadri constant of O(1)\mathcal{O}(1) at the point set (the infimum of (H⋅C)/multPC)(H\cdot C)/\mathrm{mult}_P C) over curves CC through PP), so for k≥9k\ge 9 very general points the predicted value τ=2k\tau=2\sqrt{k} is exactly Nagata's conjecture on Seshadri constants, which predicts ε(OP2(1);P)=1/k\varepsilon(\mathcal{O}_{\mathbb{P}^2}(1);P)=1/\sqrt{k} and is proven only when k\sqrt{k} is an integer. For k=9k=9 one has τ=τ0=6\tau=\tau_0=6 even though no single rational curve attains it: a Cremona-transformed sequence of lines approaches the value. The same circle of ideas powers the splitting criteria of the source paper (indecomposable EE forces the family of minimal rational curves of P(E)\mathbb{P}(E) to contain no complete curves through a general point) and the classification of rank-two Fano bundles completed in its sequel.

Problem Statement

Let XX be a complex Fano manifold of Picard number one, Pic(X)=Z⋅[HX]\mathrm{Pic}(X)=\mathbb{Z}\cdot[H_X], of index iXi_X, and let EE be a normalized rank-two vector bundle on XX, i.e. det⁡E≃OX(c1HX)\det E\simeq\mathcal{O}_X(c_1H_X) with c1∈{0,−1}c_1\in\{0,-1\}. On the Grothendieck projectivization P(E)\mathbb{P}(E) (parametrizing one-dimensional quotients of EE), let HH be the pullback of HXH_X, let LL be the tautological divisor, and let −Krel=2L−c1H-K_{\mathrm{rel}}=2L-c_1H be the relative anticanonical divisor. Define τ(E)=the unique t∈R such that −Krel+tH is nef but not ample,\tau(E)=\text{the unique }t\in\mathbb{R}\text{ such that }-K_{\mathrm{rel}}+tH\text{ is nef but not ample}, (nef means intersecting every curve nonnegatively) and τ0(E)=sup⁡{τ(ℓ)∣ℓ⊂X a rational curve},\tau_0(E)=\sup\{\tau(\ell)\mid \ell\subset X\text{ a rational curve}\}, where for a rational curve ℓ⊂X\ell\subset X with normalization ℓ~≃P1\tilde\ell\simeq\mathbb{P}^1 and splitting type E∣ℓ~≃O(a)⊕O(b)E|_{\tilde\ell}\simeq\mathcal{O}(a)\oplus\mathcal{O}(b) (the degrees of the line-bundle summands on ℓ~\tilde\ell) one sets τ(ℓ):=∣b−a∣/(HX⋅ℓ)\tau(\ell):=|b-a|/(H_X\cdot\ell); this is the slope of the minimal section of P(E)\mathbb{P}(E) over ℓ\ell, i.e. of the section of P(E)→X\mathbb{P}(E)\to X over ℓ\ell on which LL has minimal degree. Since rational curves occur among the curves computing τ\tau, one always has τ0(E)≤τ(E)\tau_0(E)\le\tau(E). Is it true that τ(E)=τ0(E)\tau(E)=\tau_0(E) for every such pair (X,E)(X,E)? Equivalently, the problem is settled either by a proof that the inequality τ0(E)≤τ(E)\tau_0(E)\le\tau(E) is always an equality, or by an explicit pair (X,E)(X,E) with τ(E)>τ0(E)\tau(E)>\tau_0(E), i.e. a rank-two vector bundle on a Fano manifold of Picard number one whose nef-cone slope strictly exceeds the supremum of the slopes computed on rational curves. Note that the equality asked for is an equality of suprema: the source paper's Example 2.13 shows τ=τ0\tau=\tau_0 may hold without any single rational curve attaining τ\tau.

References

  1. Roberto Muñoz, Gianluca Occhetta, Luis E. Solá Conde, 'On rank 2 vector bundles on Fano manifolds', Kyoto Journal of Mathematics 54 (2014), no. 1, 167–197. DOI: 10.1215/21562261-2400310; arXiv:1104.1490. https://arxiv.org/abs/1104.1490
  2. Roberto Muñoz, Gianluca Occhetta, Luis E. Solá Conde, 'A classification theorem on Fano bundles', Annales de l'Institut Fourier 64 (2014), no. 1, 341–373. DOI: 10.5802/aif.2850; arXiv:1204.4793. https://arxiv.org/abs/1204.4793
  3. Snehajit Misra, Nabanita Ray, 'Nef cones of projective bundles over surfaces and Seshadri constants', 2019, arXiv:1904.02335. https://arxiv.org/abs/1904.02335
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