Picard Rank Research Articles

Critical k-very ampleness for abelian surfaces

Let $(S,L)$ be a polarized abelian surface of Picard rank one and let $\phi$ be the function which takes each ample line bundle $L'$ to the least integer $k$ such that $L'$ is $k$-very ample but not $(k+1)$-very ample. We use Bridgeland's stability conditions and Fourier-Mukai techniques to give a closed formula for $\phi(L^n)$ as a function of $n$ showing that it is linear in $n$ for $n>1$. As a byproduct, we calculate the walls in the Bridgeland stability space for certain Chern characters.

Proof of the gamma conjecture for Fano 3-folds of Picard rank 1

We verify the (first) gamma conjecture, which relates the gamma class of a Fano variety to the asymptotics at infinity of the Frobenius solutions of its associated quantum differential equation, for all 17 of the deformation classes of Fano 3-folds of rank 1. This involves computing the corresponding limits (`Frobenius limits') for the Picard–Fuchs differential equations of Apéry type associated by mirror symmetry with the Fano families, and is achieved using two methods, one combinatorial and the other using the modular properties of the differential equations. The gamma conjecture for Fano 3-folds always contains a rational multiple of the number . We present numerical evidence suggesting that higher Frobenius limits of Apéry-like differential equations may be related to multiple zeta values.

Fourier–Mukai transforms and Bridgeland stability conditions on abelian threefolds II

We show that the conjectural construction proposed by Bayer, Bertram, Macrí and Toda gives rise to Bridgeland stability conditions for a principally polarized abelian threefold with Picard rank one by proving that tilt stable objects satisfy the strong Bogomolov–Gieseker (BG) type inequality. This is done by showing certain Fourier–Mukai transforms (FMTs) give equivalences of abelian categories which are double tilts of coherent sheaves.

On Beauville's Conjectural Weak Splitting Property

We investigate Beauville's conjecture on the Chow ring of irreducible symplectic varieties. For special irreducible symplectic varieties, we relate it to a conjecture on the existence of rational Lagrangian fibrations, which proves Beauville's conjecture in many new cases. We further apply the same techniques to reduce Beauville's conjecture to Picard rank two.

Pencils and nets of small degree on curves on smooth, projective surfaces of Picard rank 1 and very ample generator

Let S be a smooth, projective surface of Picard rank 1 and very ample generator embedding S into \({\mathbb{P}^n}\). Let \({C \in |\mathcal{O}_S(m)|}\) for \({m \geq 5}\) be a smooth curve. We prove that any base-point free, complete \({g^r_d}\) on C for \({r \in \{1,2\}}\) and d small enough is cut out by a hyperplane section restricted to a multisecant (n − r − 1)-plane.

Examples of cylindrical Fano fourfolds

We construct four different families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form $$Z\,{\times }\,{\mathbb {A}}^{1}$$ , where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective $${\mathbb {G}}_{{\text {a}}}$$ -actions. Similar constructions of cylindrical Fano threefolds were done previously in the papers by Kishimoto et al. (Affine Algebraic Geometry. CRM Proceedings & Lecture Notes, vol 54, pp 123–163, 2011; Osaka J Math 51(4):1093–1112, 2014).

Quantum reconstruction for Fano bundles on projective space

AbstractWe present a reconstruction theorem for Fano vector bundles on projective space which recovers the small quantum cohomology for the projectivization of the bundle from a small number of low-degree Gromov-Witten invariants. We provide an extended example in which we calculate the quantum cohomology of a certain Fano 9-fold and deduce from this, using the quantum Lefschetz theorem, the quantum period sequence for a Fano 3-fold of Picard rank 2 and degree 24. This example is new, and is important for the Fanosearch program.

Quantum reconstruction for Fano bundles on projective space

AbstractWe present a reconstruction theorem for Fano vector bundles on projective space which recovers the small quantum cohomology for the projectivization of the bundle from a small number of low-degree Gromov-Witten invariants. We provide an extended example in which we calculate the quantum cohomology of a certain Fano 9-fold and deduce from this, using the quantum Lefschetz theorem, the quantum period sequence for a Fano 3-fold of Picard rank 2 and degree 24. This example is new, and is important for the Fanosearch program.

The Picard rank conjecture for the Hurwitz spaces of degree up to five

We prove that the rational Picard group of the simple Hurwitz space ${\mathcal H}_{d,g}$ is trivial for $d$ up to five. We also relate the rational Picard groups of the Hurwitz spaces to the rational Picard groups of the Severi varieties of nodal curves on Hirzebruch surfaces.

Ergodic complex structures on hyperkähler manifolds

Let M be a compact complex manifold. The corresponding Teichmüller space Teich is the space of all complex structures on M up to the action of the group Diff0(M) of isotopies. The mapping class group Γ:=Diff(M)/Diff0(M) acts on Teich in a natural way. An ergodic complex structure is a complex structure with a Γ-orbit dense in Teich. Let M be a complex torus of complex dimension ≥2 or a hyperkähler manifold with b2>3. We prove that M is ergodic, unless M has maximal Picard rank (there are countably many such M). This is used to show that all hyperkähler manifolds are Kobayashi non-hyperbolic.

Supersingular K3 surfaces for large primes

Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height) then its Picard rank is 22. Along with work of Nygaard-Ogus, this conjecture implies the Tate conjecture for K3 surfaces over finite fields with p \geq 5. We prove Artin's conjecture under the additional assumption that X has a polarization of degree 2d with p > 2d+4. Assuming semistable reduction for surfaces in characteristic p, we can improve the main result to K3 surfaces which admit a polarization of degree prime-to-p when p \geq 5. The argument uses Borcherds' construction of automorphic forms on O(2,n) to construct ample divisors on the moduli space. We also establish finite-characteristic versions of the positivity of the Hodge bundle and the Kulikov-Pinkham-Persson classification of K3 degenerations. In the appendix by A. Snowden, a compatibility statement is proven between Clifford constructions and integral p-adic comparison functors.

Supersingular K3 surfaces are unirational

We show that supersingular K3 surfaces in characteristic p ≥ 5a re related by purely inseparable isogenies. This implies that they are unirational, which proves conjectures of Artin, Rudakov, Shafarevich, and Shioda. As a byproduct, we exhibit the moduli space of rigidified K3 crystals as an iterated P 1 -bundle over F p2. To complete the picture, we also establish Shioda-Inose type isogeny theorems for K3 surfaces with Picard rank ρ ≥ 19 in positive characteristic. Mathematics Subject Classification 14J28 · 14G17 · 14M20 · 14D22

Kobayashi pseudometric on hyperkähler manifolds

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincar\'e disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture for any hyperk\"ahler manifold that admits a deformation with two Lagrangian fibrations and whose Picard rank is not maximal. The Strominger-Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperk\"ahler manifolds are semi-ample. We prove that the Kobayashi pseudometric vanishes for any hyperk\"ahler manifold with $b_2\geq 13$ if the SYZ conjecture holds for all its deformations. This proves the Kobayashi conjecture for all K3 surfaces and their Hilbert schemes.

Projectivity and birational geometry of Bridgeland moduli spaces

We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wall-crossing for Bridgeland-stability and the minimal model program for the moduli space. We give three applications of our method for classical moduli spaces of sheaves on a K3 surface: 1. We obtain a region in the ample cone in the moduli space of Gieseker-stable sheaves only depending on the lattice of the K3. 2. We determine the nef cone of the Hilbert scheme of n points on a K3 surface of Picard rank one when n is large compared to the genus. 3. We verify the “Hassett-Tschinkel/Huybrechts/Sawon” conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases.

Examples of surfaces with real multiplication

AbstractWe construct explicit $K3$ surfaces over $\mathbb{Q}$ having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces constructed.

Canonical vector heights on $K3$ surfaces – A nonexistence result

A. Baragar introduced a canonical vector height on a K3 surface X defined over a number field, and showed its existence if X has Picard rank two with infinite automorphism group. In another paper, A. Baragar and R. van Lujik performed numerical computation on certain K3 surfaces with Picard rank three, which strongly suggests that, in general, a canonical vector height does not exist. In this note, we prove this last assertion. We compare the set of periodic points of one automorphism with another on certain K3 surfaces.

Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models

We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau–Ginzburg model coming from a toric degeneration.

Exceptional sequences on rational $${\mathbb{C}^{*}}$$ -surfaces

Inspired by Bondal's conjecture, we study the behavior of exceptional sequences of line bundles on rational C*-surfaces under homogeneous degenerations. In particular, we provide a sufficient criterion for such a sequence to remain exceptional under a given degeneration. We apply our results to show that, for toric surfaces of Picard rank 3 or 4, all full exceptional sequences of line bundles may be constructed via augmentation. We also discuss how our techniques may be used to construct noncommutative deformations of derived categories.

Kummer surfaces and the computation of the Picard group

AbstractWe test R. van Luijk’s method for computing the Picard group of a K3 surface. The examples considered are the resolutions of Kummer quartics in ℙ3. Using the theory of abelian varieties, the Picard group may be computed directly in this case. Our experiments show that the upper bounds provided by van Luijk’s method are sharp when sufficiently many primes are used. In fact, there are a lot of primes that yield a value close to the exact one. However, for many but not all Kummer surfaces V of Picard rank 18, we have ${\rm rk}\,{\rm Pic}(V_{\overline {\mathbb F}_{\hspace *{-.8pt}p}}) \geq 20$ for a set of primes of density at least 1/2.

The Picard group of aK3 surface and its reduction modulop

We present a method to compute the geometric Picard rank of a K3 surface over . Contrary to a widely held belief, we show that it is possible to verify Picard rank 1 using reduction at a single prime.