In algebraic geometry, the scheme-theoretic intersection of closed subschemes X, Y of a scheme W is X × W Y {\displaystyle X\times _{W}Y} , the fiber...

which is V ( x 2 − y ) {\displaystyle V(x^{2}-y)} . Their scheme-theoretic intersection is defined by the ideal ( y ) + ( x 2 − y ) = ( x 2 , y ) {\displaystyle...

products in these categories can be considered intersections (e.g. the scheme-theoretic intersection), given the objects are subobjects of the fixed...

subschemes of pure dimension. Let Z be a proper component of the scheme-theoretic intersection V × X W {\displaystyle V\times _{X}W} , that is, an irreducible...

Z_{i}=\deg(X)\deg(H)} where Zi are the irreducible components of the scheme-theoretic intersection of X and H with multiplicity (length of the local ring) mi....

concerns dimension and smoothness of scheme-theoretic intersection after some perturbation of factors in the intersection. Precisely, it states: given a connected...

assumption on regularity, the inequality can fail; see scheme-theoretic intersection#Proper intersection. Serre gives the following proof of the inequality...

taking just the set-theoretic intersection V ∩ W of the cycles in question. If the two cycles are in "good position" then the intersection product, denoted...

relative cotangent complex. derived scheme (derived tensor product gives a derived version of a scheme-theoretic intersection.) Hinich, Vladimir (1997-02-11)...

tangent spaces being in general position at intersection points). The intersection may be scheme-theoretic, in other words here the homogeneous ideal generated...

of schemes. If X and Z are closed subschemes of a scheme Y, then the fiber product X ×Y Z is exactly the intersection X ∩ Z, with its natural scheme structure...

in intersection theory: given a pair of closed subschemes V , X {\displaystyle V,X} in some ambient space, while the scheme-theoretic intersection V ∩...

Example: Let M be a complex symplectic manifold. Then the (scheme-theoretic) intersection of Lagrangian submanifolds of M carries a canonical symmetric...

X_{3}\subset \mathbb {P} ^{3}} be three surfaces. Suppose the scheme-theoretic intersection ⋂ X i {\displaystyle \bigcap X_{i}} is the disjoint union of...

higher Tor vanish, the scheme-theoretic intersection (i.e., fiber product of immersions) does not yield the correct intersection number. In the derived...

For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted;...

{\displaystyle Z(YW-Z^{2})-W(XW-YZ)} , but not a scheme-theoretic or ideal-theoretic complete intersection; meaning to say that the ideal of the variety...

divisors on it. Let Z ⊂ X {\displaystyle Z\subset X} be the scheme-theoretic intersection of A + D {\displaystyle A+D} and B + D {\displaystyle B+D} (viewing...

cographs. However, a geometric intersection graph representation does not always imply the existence of an adjacency labeling scheme, because it may require...

algorithmic number theory, is the study of algorithms for performing number theoretic computations. The best known problem in the field is integer factorization...

associated partial ordering. Historically, filters generalized to order-theoretic lattices before arbitrary partial orders. In the case of lattices, downward...

minimal field of definition. One disadvantage of the scheme-theoretic definition is that a scheme over k cannot have an L-valued point if L is not an extension...

In algebraic geometry, a derived scheme is a homotopy-theoretic generalization of a scheme in which classical commutative rings are replaced with derived...

calculating the planes' point of intersection and then taking a specified coordinate of that intersection. Blakley's scheme is less space-efficient than Shamir's;...

{\displaystyle |D|} (as a set, at least: there may be more subtle scheme-theoretic considerations as to what the structure sheaf of Bl {\displaystyle...

n {\displaystyle n} -element set; two vertices are adjacent when the intersection of the two vertices (subsets) contains ( k − 1 ) {\displaystyle (k-1)}...

Covariance intersection (CI) is an algorithm for combining two or more estimates of state variables in a Kalman filter when the correlation between them...

is allowed to carry non-trivial automorphisms and consequently intersection-theoretic operations must take this into account. For example, the degree...

{\displaystyle {\mathcal {F}}} . If X is a flat, locally complete intersection (for example, regular) scheme over a henselian discrete valuation ring, then the constant...

with coefficients called intersection numbers. For any subvarieties Y {\displaystyle Y} and Z {\displaystyle Z} of a smooth scheme X {\displaystyle X} over...