of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related...

a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at...

called semi-ample if some positive tensor power L ⊗ a {\displaystyle L^{\otimes a}} is basepoint-free. It follows that a semi-ample line bundle is nef. Semi-ample...

anticanonical bundle is the corresponding inverse bundle ω − 1 {\displaystyle \omega ^{-1}} . When the anticanonical bundle of V {\displaystyle V} is ample, V {\displaystyle...

tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact...

{\displaystyle i^{*}x_{0},\ldots ,i^{*}x_{n}} . Conversely, given an ample line bundle L → X {\displaystyle {\mathcal {L}}\to X} globally generated by n...

interpretation of the ample line bundle over the moduli space of vector bundles on a compact Riemann surface, known as the Quillen determinant line bundle. It can be...

geometry. For example, the fact that the canonical bundle is a negative multiple of the ample line bundle O ( 1 ) {\displaystyle {\mathcal {O}}(1)} means...

third power of an ample line bundle is normally generated. The Mumford–Kempf theorem states that the fourth power of an ample line bundle is quadratically...

Kodaira embedding theorem claims that a positive line bundle is ample, and conversely, any ample line bundle admits a Hermitian metric with − 1 Θ {\displaystyle...

computing the Hilbert polynomial of line bundles on a curve. If a line bundle L {\displaystyle {\mathcal {L}}} is ample, then the Hilbert polynomial will...

varieties, and if L is a big line bundle on X, then f*L is a big line bundle on Y. All ample line bundles are big. Big line bundles need not determine birational...

that if L is a big nef line bundle (for example, an ample line bundle) on a complex projective manifold with canonical line bundle K, then the coherent...

there exists a very ample sheaf on X relative to S. Indeed, if X is proper, then an immersion corresponding to the very ample line bundle is necessarily closed...

In algebraic geometry, given an ample line bundle L on a compact complex manifold X, Matsusaka's big theorem gives an integer m, depending only on the...

Temple Seshadri constant, in algebraic geometry is an invariant of an ample line bundle L at a point P on an algebraic variety Seshadripuram, residential...

canonical line bundle makes projective spaces prime examples of Fano varieties, equivalently, their anticanonical line bundle is ample (in fact very ample). Their...

complete toric variety that has no non-trivial line bundle; thus, in particular, it has no ample line bundle. Definition 1.1.12 in Ginzburg, V., 1998. Lectures...

K3 surface together with an ample line bundle L such that L is primitive (that is, not 2 or more times another line bundle) and c 1 ( L ) 2 = 2 g − 2 {\displaystyle...

linearizations of the trivial line bundle. See Example 2.16 of [1] for an example of a variety for which most line bundles are not linearizable. Given an...

In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle L at a point P on an algebraic variety. It was introduced by Demailly...

geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample. Let D be a nef divisor on a smooth projective surface...

the theory of schemes, a related notion is ample line bundle. (For example, if L is an ample line bundle, some power of it is generated by global sections...

divisorial scheme is a scheme admitting an ample family of line bundles, as opposed to an ample line bundle. In particular, a quasi-projective variety...

zero, L {\displaystyle L} is an ample line bundle on X {\displaystyle X} , and K X {\displaystyle K_{X}} a canonical bundle, then H j ( X , K X ⊗ L ) = 0...

{\displaystyle X} is a smooth algebraic surface and L {\displaystyle L} is an ample line bundle on X {\displaystyle X} of degree d {\displaystyle d} , then for sufficiently...

projective variety of dimension n ≥ 1 over a field, and let L be an ample line bundle on X. Then the section ring of L R = ⨁ j ≥ 0 H 0 ( X , L j ) {\displaystyle...

Hilbert polynomial Φ {\displaystyle \Phi } . For a relatively very ample line bundle L ∈ Pic ( X ) {\displaystyle {\mathcal {L}}\in {\text{Pic}}(X)} and...

appear as standard forms for varieties without a minimal model. Ample line bundle Fiber bundle Fibration Quasi-fibration Matsuki, Kenji (2002), Introduction...

Lefschetz proved that the line bundle L {\displaystyle L} , associated to the Hermitian form H {\displaystyle H} is ample if and only if H {\displaystyle...