Projective morphisms before Proj

Depends on

• Relative projective space from standard charts
• Closed immersions of schemes

Used by

• Every smooth proper curve admits a projective embedding Corollary
• A proper nonprojective scheme from glued projective spaces Counterexample
• The intersection form is not negative definite on all divisor classes Counterexample
• The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
• Exceptional curves of the first kind and their contractions Definition
• Hilbert function and Euler characteristic on a projective scheme Definition
• Intersection numbers of Cartier divisors on a smooth projective surface Definition
• Projectivity via a coherent projective bundle Definition
• Quasi-projective morphisms before Proj Definition
• The canonical divisor of a smooth projective surface Definition
• Ramification of the double cover y²=f(x) Example
• The empty morphism is finite, proper and projective Example
• A projective morphism has a relative Proj presentation Lemma
• A vector bundle on the projective line has a line subbundle of maximal degree Lemma
• Blowups of integral projective complex schemes are projective Lemma
• Chow lemma for proper Noetherian schemes Lemma
• Eventual generation of coherent projective twists Lemma
• Existence and biduality from a projective embedding Lemma
• Finite schemes over projective schemes are projective over a Noetherian affine base Lemma
• Large ample twists of a line bundle are very ample Lemma
• Semistable loci of finite-quiver representations are open Lemma
• SNC projective compactifications of smooth quasiprojective varieties Lemma
• The Picard group of a point blowup of the projective plane Lemma
• The product of two projective lines is an integral smooth projective surface Lemma
• Projective and proper are distinct notions Remark
• Blowups of finite type ideals are locally H-projective, and proper Theorem
• Coherent higher direct images under proper morphisms Theorem
• Degree of the coherent Hilbert polynomial Theorem
• Euler characteristic is a Hilbert polynomial Theorem
• Grothendieck-Riemann-Roch for projective morphisms Theorem
• Negativity of contracted curves on regular surfaces Theorem
• Projective morphisms are proper Theorem
• Serre duality for coherent sheaves on a smooth proper curve, Ext form Theorem
• Serre duality for coherent sheaves on projective space Theorem
• Serre vanishing for coherent sheaves and ample twists Theorem
• Smooth projective proper hyperresolutions Theorem
• The surface intersection product is symmetric and bilinear Theorem
• Vanishing of H¹ in a fixed ample direction Theorem

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

• Stacks Project, Morphisms of Schemes, §29.44 Definition 29.44.1(2) (standard reference, not scraped)