Integral schemes

Depends on

• Schemes
• The reduction of a scheme
• Irreducible topological spaces and irreducible subsets in the subspace topology
• Affine open subschemes

Used by

• A degree-zero line bundle with a nonzero section is trivial Corollary
• Blowing up a nonzero ideal on an integral scheme is birational Corollary
• Degree is additive on invertible sheaves over a proper curve Corollary
• Every smooth proper curve admits a projective embedding Corollary
• Normalization resolves the singularities of a projective curve Corollary
• Proper birational normal curves agree off finitely many points Corollary
• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• The genus relation for unramified covers of curves Corollary
• A Weil divisor that is not Cartier at the vertex of the quadric cone Counterexample
• Finite normalization alone does not make a curve regular Counterexample
• Smoothness of the source cannot be dropped in the extension of rational maps Counterexample
• The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
• The maximal-contact mechanism fails in positive characteristic Counterexample
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
• Algebraic cycles and the cycle group of a scheme of finite type over a field Definition
• Base points and base-point-free linear systems Definition
• Birational morphisms of integral finite-type schemes Definition
• Canonical bundle and canonical divisors Definition
• Curves over a field Definition
• Degree divisor proper curve Definition
• Exceptional curves of the first kind and their contractions Definition
• Geometric genus of a singular curve Definition
• Geometric properties of fibres Definition
• Intersection multiplicity of closed subschemes at a point Definition
• Intersection numbers of Cartier divisors on a smooth projective surface Definition
• Invariance of the blowup under invertible (fractional) rescaling of the ideal Definition
• Locally factorial scheme Definition
• Normal scheme modifications and normalized point blowups Definition
• Order codimension one rational function Definition
• Principal weil divisor and class group Definition
• Rational equivalence and the Chow group of cycles Definition
• Rational maps of integral finite-type schemes Definition
• Rational section line bundle Definition
• Scheme-theoretic varieties Definition
• Sheaf total quotient rings Definition
• The canonical divisor of a smooth projective surface Definition
• Weil divisor normal noetherian scheme Definition
• Divisor of a rational function on the projective line Example
• Principal divisors on the projective line have degree zero Example
• Pulling a divisor back along the cusp normalization Example

…and 74 more results.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• The Stacks Project, Schemes, Section 12 (standard reference, not scraped)