Ample Line Bundle Research Articles

Seshadri constants on blow‐ups of Hirzebruch surfaces

AbstractLet be integers and let denote the Hirzebruch surface with invariant . We compute the Seshadri constants of an ample line bundle at an arbitrary point of the ‐point blow‐up of when and at a very general point when or . We also discuss several conjectures on linear systems of curves on the blow‐up of at very general points.

Non-Density of the Exceptional Components of the Noether–Lefschetz Locus

Abstract We study when the Picard group of smooth surfaces of degree $d\geq 5$ in $\mathbb{P}^{3}$ acquires extra classes. In particular we show that the so-called exceptional components of the Noether–Lefschetz locus are not Zariski dense. This answers a 1991 question of C. Voisin. We also obtain similar results for the Noether–Lefschetz locus for suitable $(Y,L)$, where $Y$ is a smooth projective three-fold and $L$ a very ample line bundle. Both results are applications of the Zilber–Pink viewpoint recently developed by the authors for arbitrary (polarized, integral) variations of Hodge structures.

Crepant Resolutions of Stratified Varieties via Gluing

Abstract Let $X$ be a variety with a stratification ${\mathcal{S}}$ into smooth locally closed subvarieties such that $X$ is locally a product along each stratum (e.g., a symplectic singularity). We prove that assigning to each open subset $U \subset X$ the set of isomorphism classes of locally projective crepant resolutions of $U$ defines an ${\mathcal{S}}$-constructible sheaf of sets. Thus, for each stratum $S$ and basepoint $s \in S$, the fundamental group acts on the set of germs of projective crepant resolutions at $s$, leaving invariant the germs extending to the entire stratum. Global locally projective crepant resolutions correspond to compatible such choices for all strata. For example, if the local projective crepant resolutions are unique, they automatically glue uniquely. We give criteria for a locally projective crepant resolution $\rho : \tilde X \to X$ to be globally projective. We show that the sheafification of the presheaf $U \mapsto \operatorname{Pic}(\rho ^{-1}(U)/U)$ of relative Picard classes is also constructible. The resolution is globally projective only if there exist local relatively ample bundles whose classes glue to a global section of this sheaf. The obstruction to lifting this section to a global ample line bundle is encoded by a gerbe on the singularity $X$. We show the gerbes are automatically trivial if $X$ is a symplectic quotient singularity. Our main results hold in the more general setting of partial crepant resolutions, that need not have smooth source. We apply the theory to symmetric powers and Hilbert schemes of surfaces with du Val singularities, finite quotients of tori, multiplicative, and Nakajima quiver varieties, as well as to canonical three-fold singularities.

Ng\^o support theorem and polarizability of quasi-projective commutative group schemes

We prove that any commutative group scheme over an arbitrary base scheme of finite type over a field with connected fibers and admitting a relatively ample line bundle is polarizable in the sense of Ng\^o. This extends the applicability of Ng\^o's support theorem to new cases, for example to Lagrangian fibrations with integral fibers and has consequences to the construction of algebraic classes.

Line bundles on the moduli space of Lie algebroid connections over a curve

We explore algebro-geometric properties of the moduli space of holomorphic Lie algebroid (L) connections on a compact Riemann surface X of genus g≥3. A smooth compactification of the moduli space of L-connections, such that underlying vector bundle is stable, is constructed; the complement of the moduli space inside the compactification is a divisor. A criterion for the numerical effectiveness of the boundary divisor is given. We compute the Picard group of the moduli space, and analyze Lie algebroid Atiyah bundles associated with an ample line bundle. This enables us to conclude that regular functions on the space of certain Lie algebroid connections are constants. Moreover, under some condition, it is shown that the moduli space of L-connections does not admit non-constant algebraic functions. Rationally connectedness of the moduli spaces is explored.

Normal generation of special line bundles on multiple coverings

Let X be a smooth curve of genus g which is a multiple covering ϕ : X → Y of a smooth curve Y. We define an invariant Δ ϕ of the map ϕ , which is the maximum among the integers m such that for a base point free line bundle L on X with deg L ≤ g − 1 the inequality Cliff ( L ) ≤ m implies that L is composed with ϕ . Our main result is that a very ample special line bundle L on X with deg L > 3 g − 3 2 is normally generated when Cliff ( L ) ≤ Δ ϕ under additional numerical conditions.

Degree of the 3-secant variety

In this paper, we present a formula for the degree of the 3-secant variety of a nonsingular projective variety embedded by a 5-very ample line bundle. The formula is provided in terms of Segre classes of the tangent bundle of a given variety. We use the generalized version of the double point formula to reduce the calculation into the case of the 2-secant variety. As a resolution of the 2-secant variety, we use secant bundle and compute powers of desired algebraic cycles.

On the zero set of the holomorphic sectional curvature

AbstractA notable example due to Heier, Lu, Wong, and Zheng shows that there exist compact complex Kähler manifolds with ample canonical line bundle such that the holomorphic sectional curvature is negative semi‐definite and vanishes along high‐dimensional linear subspaces in every tangent space. The main result of this note is an upper bound for the dimensions of these subspaces. Due to the holomorphic sectional curvature being a real‐valued bihomogeneous polynomial of bidegree (2,2) on every tangent space, the proof is based on making a connection with the work of D'Angelo on complex subvarieties of real algebraic varieties and the decomposition of polynomials into differences of squares. Our bound involves an invariant that we call the holomorphic sectional curvature square decomposition length, and our arguments work as long as the holomorphic sectional curvature is semi‐definite, be it negative or positive.

On rank 3 quadratic equations of projective varieties

Let X ⊂ P r X \subset \mathbb {P}^r be a linearly normal variety defined by a very ample line bundle L L on a projective variety X X . Recently it is shown by Kangjin Han, Wanseok Lee, Hyunsuk Moon, and Euisung Park [Compos. Math. 157 (2021), pp. 2001–2025] that there are many cases where ( X , L ) (X,L) satisfies property Q R ( 3 ) \mathsf {QR} (3) in the sense that the homogeneous ideal I ( X , L ) I(X,L) of X X is generated by quadratic polynomials of rank 3 3 . The locus Φ 3 ( X , L ) \Phi _3 (X,L) of rank 3 3 quadratic equations of X X in P ( I ( X , L ) 2 ) \mathbb {P}\left ( I(X,L)_2 \right ) is a projective algebraic set, and property Q R ( 3 ) \mathsf {QR} (3) of ( X , L ) (X,L) is equivalent to that Φ 3 ( X ) \Phi _3 (X) is nondegenerate in P ( I ( X ) 2 ) \mathbb {P}\left ( I(X)_2 \right ) . In this paper we study geometric structures of Φ 3 ( X , L ) \Phi _3 (X,L) such as its minimal irreducible decomposition. Let Σ ( X , L ) = { ( A , B ) ∣ A , B ∈ P i c ( X ) , L = A 2 ⊗ B , h 0 ( X , A ) ≥ 2 , h 0 ( X , B ) ≥ 1 } . \begin{equation*} \Sigma (X,L) \!=\! \{ (A,B) \mid A,B \!\in \! {Pic}(X),~L \!=\! A^2 \otimes B,~h^0 (X,A) \!\geq \! 2,~h^0 (X,B) \!\geq \! 1 \}. \end{equation*} We first construct a projective subvariety W ( A , B ) ⊂ Φ 3 ( X , L ) W(A,B) \subset \Phi _3 (X,L) for each ( A , B ) (A,B) in Σ ( X , L ) \Sigma (X,L) . Then we prove that the equality Φ 3 ( X , L ) = ⋃ ( A , B ) ∈ Σ ( X , L ) W ( A , B ) \begin{equation*} \Phi _3 (X,L) ~=~ \bigcup _{(A,B) \in \Sigma (X,L)} W(A,B) \end{equation*} holds when X X is locally factorial. Thus this is an irreducible decomposition of Φ 3 ( X , L ) \Phi _3 (X,L) when P i c ( X ) {Pic} (X) is finitely generated and hence Σ ( X , L ) \Sigma (X,L) is a finite set. Also we find a condition that the above irreducible decomposition is minimal. For example, it is a minimal irreducible decomposition of Φ 3 ( X , L ) \Phi _3 (X,L) if P i c ( X ) {Pic}(X) is generated by a very ample line bundle.

Chebyshev potentials, Fubini–Study metrics, and geometry of the space of Kähler metrics

AbstractThe Chebyshev potential of a Hermitian metric on an ample line bundle over a projective variety, introduced by Witt Nyström, is a convex function defined on the Okounkov body. It is a generalization of the symplectic potential of a torus‐invariant Kähler potential on a toric variety, introduced by Guillemin, that is a convex function on the Delzant polytope. A folklore conjecture asserts that a curve of Chebyshev potentials associated to a subgeodesic in the space of positively curved Hermitian metrics is linear in the time variable if and only if the subgeodesic is a geodesic in the Mabuchi metric. This is classically true in the special toric setting, and in general Witt Nyström established the sufficiency. The main obstacle in the conjecture is that it is difficult to compute Chebyshev potentials, that are currently only known on the Riemann sphere and toric varieties. The goal of this article is to disprove this conjecture. To that end we characterize the geodesics consisting of Fubini–Study metrics for which the conjecture is true on the hyperplane bundle of the projective space. The proof involves explicitly solving the Monge–Ampère equation describing geodesics on the subspace of Fubini–Study metrics and computing their Chebyshev potentials.

Syzygies of Kummer varieties

We study syzygies of Kummer varieties proving that their behavior is half of the abelian varieties case. Namely, an m m th power of an ample line bundle on a Kummer variety satisfies the Green-Lazarsfeld property ( N p ) (N_p) , if m > p + 2 2 m > \frac {p+2}{2} .

Ein–Lazarsfeld–Mustopa conjecture for the blow-up of a projective space

We solve the Ein–Lazarsfeld–Mustopa conjecture for the blow up of a projective space along a linear subspace. More precisely, let X be the blow up of {mathbb {P}}^n at a linear subspace and let L be any ample line bundle on X. We show that the syzygy bundle M_{L} defined as the kernel of the evalution map H^{0}(X,L)otimes {mathcal {O}}_{X}rightarrow L is L-stable. In the last part of this note we focus on the rigidness of M_{L} to study the local shape of the moduli space around the point [M_{L}].

Integral Points on Varieties With Infinite Étale Fundamental Group

Abstract We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $F$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $\varphi \colon Y \to X$ of degree at least $2\dim (X)^{2}$, there exists an ample line bundle $\mathscr{L}$ on $Y$ such that for a general member $D$ of the complete linear system $|\mathscr{L}|$, $D$ is geometrically irreducible and any set of $\varphi (D)$-integral points on $X$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable.

Ample line bundles and generation time

AbstractWe prove that ifXis a regular quasi-projective variety of dimensiond, the set of line bundles{𝒪X⁢(n)}n∈ℤ{\{\mathcal{O}_{X}(n)\}_{n\in{\mathbb{Z}}}}generates the bounded derived category ofXindsteps. This proves new cases of a conjecture of Orlov as well as a conjecture of Elagin and Lunts.

The Demailly system for a direct sum of ample line bundles on Riemann surfaces

The Demailly system for a direct sum of ample line bundles on Riemann surfaces

Metrised ample line bundles in non-Archimedean geometry

Metrised ample line bundles in non-Archimedean geometry

Expected local topology of random complex submanifolds

Let n ≥ 2 n\geq 2 and r ∈ { 1 , ⋯ , n − 1 } r\in \{1, \cdots , n-1\} be integers, M M be a compact smooth Kähler manifold of complex dimension n n , E E be a holomorphic vector bundle with complex rank r r and equipped with a Hermitian metric h E h_E , and L L be an ample holomorphic line bundle over M M equipped with a metric h h with positive curvature form. For any d ∈ N d\in \mathbb N large enough, we equip the space of holomorphic sections H 0 ( M , E ⊗ L d ) H^0(M,E\otimes L^d) with the natural Gaussian measure associated to h E h_E , h h and its curvature form. Let U ⊂ M U\subset M be an open subset with smooth boundary. We prove that the average of the ( n − r ) (n-r) -th Betti number of the vanishing locus in U U of a random section s s of H 0 ( M , E ⊗ L d ) H^0(M,E\otimes L^d) is asymptotic to ( n − 1 r − 1 ) d n ∫ U c 1 ( L ) n {n-1 \choose r-1} d^n\int _U c_1(L)^n for large d d . On the other hand, the average of the other Betti numbers is o ( d n ) o(d^n) . The first asymptotic recovers the classical deterministic global algebraic computation. Moreover, such a discrepancy in the order of growth of these averages is new and contrasts with all known other smooth Gaussian models, in particular the real algebraic one. We prove a similar result for the affine complex Bargmann-Fock model.

Seshadri constants and K-stability of Fano manifolds

We give a lower bound of the δ-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano hypersurfaces of arbitrarily large index, as well as the uniform K-stability of most families of smooth Fano threefolds of Picard number one.

The space of finite-energy metrics over a degeneration of complex manifolds

Given a degeneration of projective complex manifolds X→𝔻 * with meromorphic singularities, and a relatively ample line bundle L on X, we study spaces of plurisubharmonic metrics on L, with particular focus on (relative) finite-energy conditions. We endow the space ℰ ^ 1 (L) of relatively maximal, relative finite-energy metrics with a d 1 -type distance given by the Lelong number at zero of the collection of fiberwise Darvas d 1 -distances. We show that this metric structure is complete and geodesic. Seeing X and L as schemes X K , L K over the discretely-valued field K=ℂ((t)) of complex Laurent series, we show that the space ℰ 1 (L K an ) of non-Archimedean finite-energy metrics over L K an embeds isometrically and geodesically into ℰ ^ 1 (L), and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case.

Sets of special subvarieties of bounded degree

Let $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base $S$, and let $\mathbb {V} = R^{2k} f_{*} \mathbb {Z}(k)$ be the integral variation of Hodge structure coming from degree $2k$ cohomology it induces. Associated to $\mathbb {V}$ one has the so-called Hodge locus$\textrm {HL}(S) \subset S$, which is a countable union of ‘special’ algebraic subvarieties of $S$ parametrizing those fibres of $\mathbb {V}$ possessing extra Hodge tensors (and so, conjecturally, those fibres of $f$ possessing extra algebraic cycles). The special subvarieties belong to a larger class of so-called weakly special subvarieties, which are subvarieties of $S$ maximal for their algebraic monodromy groups. For each positive integer $d$, we give an algorithm to compute the set of all weakly special subvarieties $Z \subset S$ of degree at most $d$ (with the degree taken relative to a choice of projective compactification $S \subset \overline {S}$ and very ample line bundle $\mathcal {L}$ on $\overline {S}$). As a corollary of our algorithm we prove conjectures of Daw–Ren and Daw–Javanpeykar–Kühne on the finiteness of sets of special and weakly special subvarieties of bounded degree.