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...al crossing singularities are not usually [[normal schemes]] (in the sense of the local rings being integrally closed). ==Normal crossing divisors== ...

...pancy''' of the pair (''X'', ''D'') measures the degree of the singularity of the pair. ...| first1=János | last2=Mori | first2=Shigefumi | title=Birational geometry of algebraic varieties | publisher=[[Cambridge University Press]] | series=Cam ...

...f there is an open affine cover <math>U_i = \operatorname{Spec} A_i</math> of ''X'' and nonzerodivisors <math>f_i \in A_i</math> such that the intersecti == An effective Cartier divisor as the zero-locus of a section of a line bundle == ...

...r the definition of [[invertible sheaf|invertible sheaves]] in [[algebraic geometry]]. ...words, <math> M_P\cong R_P </math> for all [[prime element|primes]] ''P'' of ''R''. Now, if ''M'' is an invertible ''R''-module, then its [[Duality (mat ...

...y]] ''X'' is a real number connected with the rate of growth of the number of rational points on the variety with respect to the embedding defined by the ...e closed real cone of [[effective divisor]]s in the [[Néron–Severi group]] of ''X''. If α is negative, then ''X'' is [[pseudo-canonical variety|pseudo-c ...

...here is no restriction on the [[characteristic of a field|characteristic]] of the underlying field, while the extensions require characteristic 0.<ref>{{ == Statement for hyperplane sections of smooth varieties== ...

...n terms of abstract [[algebraic geometry]] and to describe some basic uses of projective spaces. ...oted by '''k'''[''V'']. It is a naturally [[graded algebra]] by the degree of polynomials. ...

...''P''; and are helpful in specifying what should happen about the process of reducing to ''R''/''P'' when the most naive way fails to make sense. ...ducible]], [[Singular point of an algebraic variety|singular]], [[Glossary of scheme theory|non-reduced]] ) over the appropriate [[residue field]] for [[ ...

...catively closed set''' (or '''multiplicative set''') is a [[subset]] ''S'' of a [[ring (mathematics)|ring]] ''R'' such that the following two conditions ..., a multiplicative set is a [[submonoid]] of the multiplicative [[monoid]] of a ring. ...

In [[algebraic geometry]], '''Reider's theorem''' gives conditions for a [[line bundle]] on a proje ...e surface ''X''. Denote by ''K''_''X'' the [[canonical divisor]] of X. ...

{{short description|Characterization of even perfect numbers}} ...eorem on perfect numbers and Mersenne primes|the theorem on the infinitude of prime numbers|Euclid's theorem}} ...

...or more generally a subset of a variety encodes the distribution of points of given [[height function|height]]. ...ith height function ''H'', such that there are only finitely many elements of bounded height, define a ''counting function'' ...

{{Short description|Singularities of projective varieties}} ...''terminal singularities''' are special cases that appear as singularities of [[Minimal model program|minimal models]]. They were introduced by {{harvtxt ...

...> that, roughly, parametrizes ''n''-th order Taylor expansions of sections of&nbsp;''L''. ...to X</math> to <math>V(I^{n+1}) \subset X \times X</math>. Then the bundle of ''n''-th order principal parts is<ref>{{harvnb|Fulton|1998|loc=Example 2.5. ...

{{Short description|Generalization of equivalence classes to scheme theory}} ...tion''' <math>f: R \to X \times X</math> is a [[coequalizer]] for the pair of maps ...

...space is convex. A variety <math>X</math> is called convex if the pullback of the tangent bundle to a stable [[Algebraic curve|rational curve]] <math>f:C ...n \{ tp + (1-t)q : t \in [0,1] \} \cap U_p</math>. Since the vector bundle of <math>\mathbb{R}^n</math> is trivial, hence globally generated, there is a ...

...manifold]] to classes in its integral [[cohomology]]. It is the only case of the [[Hodge conjecture]] which has been proved for all Kähler manifolds.<re == Statement of the theorem == ...

888 is a [[base ten]] [[repdigit]] (a number all of whose digits are equal),<ref>{{Cite OEIS|sequencenumber=A010785|name=Repdig ...eger up to 888 itself may be represented as a sum of distinct [[divisor]]s of 888.<ref>[http://jean-paul.davalan.pagesperso-orange.fr/arit/pract/practica ...

...dered to be relatively simple among all varieties, although there are many of them. ...'' ≤ ''n'', by the [[adjunction formula]]. (In fact, a smooth hypersurface of degree ''d'' ≤ ''n'' in '''P'''ⁿ is a [[Fano variety]] and hence ...

...[[Denmark]]) is a German mathematician, known for his work in [[algebraic geometry]]. ...e for Theoretical Physics]] in [[Trieste]], Italy. He is also editor for [[Geometry & Topology]]. ...