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...osely connected with divisors (cf. [[Divisor|Divisor]]). With each Cartier divisor $ D $ is called very ample if there exists an imbedding $ \phi : X \rightarrow P ^ {N} ( k) $

A theorem stating that there is a strong restriction for the [[canonical divisor]] of an [[Algebraic variety|algebraic variety]] to be negative while the po ...cf. also [[Divisor|Divisor]]). The canonical divisor $K _ { X }$ is a Weil divisor on $X$ corresponding to a non-zero rational differential $n$-form for $n =

is the [[canonical divisor]] of the surface $ X $. on a smooth complete rational surface is ample (cf. [[Ample sheaf|Ample sheaf]]), then $ X $

A generalization of the concept of a [[Divisor|divisor]] of positive degree on a [[Riemann surface|Riemann surface]]. A holomorphi defined by a divisor of degree $ d $

...ndles) play an important part in complex analytic geometry. Each [[Divisor|divisor]] on the space $ X $ ...l analytic line bundles on a projective algebraic variety are defined by a divisor. The imbeddability of a complex space $ X $

..._ { X }$ [[#References|[a9]]], Formula I.17, in terms of the double point divisor of a sufficiently general projection of $X$ into $P^3$. ...ed to show vanishing of real cohomology groups, Kodaira showed that for an ample line bundle $L$ on a compact complex projective manifold, $h ^ { i } ( K _

is ample (cf. [[Ample sheaf|Ample sheaf]]). The basic research into such varieties was done by G. Fano ([[#Re the self-intersection index of the anti-canonical divisor $ (- K _ {X} ^ {3} ) \leq 64 $.

...{ 2 } ) \in \bf R$ for any Cartier divisor $D$ on $X$ (cf. also [[Divisor|Divisor]]; [[Intersection index (in algebraic geometry)|Intersection index (in alge Let $X$ be a normal algebraic variety and $B$ an effective $\mathbf{Q}$-divisor such that the pair $( X , B )$ is weakly log terminal (cf. [[Kawamata ratio

is ample (cf. [[Ample sheaf|Ample sheaf]]) if and only if $ \phi _ {L} $ denotes the self-intersection number. Every Abelian surface admits an ample invertible sheaf and hence is projective (cf. [[Projective scheme|Projectiv

and states that for any [[Divisor|divisor]] $ D $ is the [[canonical divisor]] and $ g $

corresponds to an effective irreducible [[Divisor|divisor]], then $ H ^ {1} ( X, D) = 0 $. is an even number for any divisor $ D $.

..._ { s }$; cf. also [[Extension of a field|Extension of a field]]). A prime divisor of $K$ is an equivalence class $\text{p}$ of absolute values (cf. also [[No ...{ simp } } ( M ) \neq \emptyset$, then $V ( M )$ is Zariski-dense in $V$). Ample fields, in particular P$S$C fields, have the nice property that the inverse

is known as the speciality index of the divisor $ D $. It has been proved that, for any divisor $ D $

...le sheaves of $ \mathcal{O}_{X} $-modules and are closely connected with [[Divisor|divisors]] on $ X $; the set of line bundles with the tensor product operat ...$ X $ into $ \mathbf{P}^{n} $ are said to be very [[Ample vector bundle|'''ample''']].

[[divisor]] $D$ on $X$, this theorem establishes the equality [[canonical divisor]] on $X$. If $n = 1$, a relation equivalent to the above was found as early