In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map f : X → Y {\displaystyle f:X\rightarrow Y} of varieties is a...

divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors...

^{1}} . This P 1 {\displaystyle \mathbf {P} ^{1}} is called the exceptional divisor, and by definition it is the projectivized normal space at P {\displaystyle...

symmetry between the two singular axes because one of them is an exceptional divisor for a previous blowup, so it is now permissible to blow up just one...

V {\displaystyle V} , and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor − K {\displaystyle K} with K {\displaystyle...

{\displaystyle \mathbb {A} ^{n+1}} , where the locus x = 0 in L is the exceptional divisor. (cf. Hartshorne, Ch. I, the end of § 4.) In general, S p e c ( Sym...

bundles. The tautological bundle, which appears for instance as the exceptional divisor of the blowing up of a smooth point is the sheaf O ( − 1 ) {\displaystyle...

^{-1}(Z)\hookrightarrow {\widetilde {X}}} is an effective Cartier divisor, called the exceptional divisor, and (2) π {\displaystyle \pi } is universal with respect...

the exceptional curve E of the blow-up π : X → Y {\displaystyle \pi \colon X\to Y} has E 2 = − 1 {\displaystyle E^{2}=-1} . Every effective divisor on...

}I^{n}\right)\to Y} be the blow-up of Y along X. Then, by definition, the exceptional divisor is the pre-image E = π − 1 ( X ) {\displaystyle E=\pi ^{-1}(X)} ;...

40). exceptional 1.  Corresponding to something of lower dimension under a birational correspondence, as in exceptional curve, exceptional divisor 2.  An...

What is being divided is called the dividend, which is divided by the divisor, and the result is called the quotient. At an elementary level the division...

K_{X}=f^{*}(K_{Y})+\sum _{i}a_{i}E_{i}} where the sum is over the irreducible exceptional divisors, and the ai are rational numbers, called the discrepancies. Then...

and let E be the exceptional curve of the blow-up π : X → P 2 {\displaystyle \pi \colon X\to \mathbb {P} ^{2}} . Then the divisor H + E is big but not...

desingularization, together with information about multiples of the exceptional divisors. This allows the complete calculation of the Chow group of 0-cycles...

(the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than the absolute value of the divisor. A fundamental...

_{1}\to \mathbb {CP} ^{2}} is the blow up and E {\displaystyle E} the exceptional divisor. The measure d σ {\displaystyle d\sigma } on the horizontal and vertical...

and E, where H is the total transform of a hyperplane and E is the exceptional divisor. The total transform of a degree 6 hypersurface is 6H, and Steiner...

{\displaystyle Z} , the blow up locus Z {\displaystyle Z} is replaced by an exceptional divisor E {\displaystyle E} and the rest of the manifold is left undisturbed...

special divisors, certain divisors on a curve C that determine more compatible functions than would be predicted. In classical language, special divisors move...

Since the exceptional divisor E := π − 1 ( Z ) ↪ X ~ {\displaystyle E:=\pi ^{-1}(Z)\hookrightarrow {\widetilde {X}}} is an effective Cartier divisor and the...

itself a power. 72 is an abundant number. With exactly twelve positive divisors, including 12 (one of only two sublime numbers), 72 is also the twelfth...

In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case. Using fraction notation...

symplectic form is naturally extended to the smooth part of the exceptional divisors of M [ n ] {\displaystyle M^{[n]}} . It is extended to the rest of...

integer n ≥ 1 {\displaystyle n\geq 1} , there exists a primitive prime divisor p {\displaystyle p} that divides a n − b n {\displaystyle a^{n}-b^{n}}...

words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general type...

along a closed subscheme X, X ~ {\displaystyle {\widetilde {X}}} the exceptional divisor and g : g ~ : X ~ → X {\displaystyle g:{\widetilde {g}}:{\widetilde...

number of elements) must be a divisor of the order of G {\displaystyle G} which is 3. Since 3 is prime, its only divisors are 1 and 3, so either H {\displaystyle...

Lagrange's theorem is not true in general: given a finite group G and a divisor d of |G|, there does not necessarily exist a subgroup of G with order d:...

singular', in the sense that the canonical divisor K X i {\displaystyle K_{X_{i}}} is no longer a Cartier divisor, so the intersection number K X i ⋅ C {\displaystyle...