Projective scheme
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| Projective scheme | |
|---|---|
| Name | Projective scheme |
| Field | Algebraic geometry |
| Introduced | 1960s |
| Introduced by | Alexander Grothendieck |
Projective scheme. A projective scheme is a fundamental object in modern algebraic geometry connecting Alexander Grothendieck's scheme theory with classical projective space constructions. It generalizes projective varieties used in the work of Bernhard Riemann, David Hilbert, and Andre Weil, and it underpins advances by Jean-Pierre Serre, Oscar Zariski, and Bernard Teissier in sheaf cohomology and intersection theory.
Definition
A projective scheme over a base Spec point such as Spec Z or Spec k is defined as a closed subscheme of a projective space P^n over that base; the notion was formalized by Grothendieck in the context of EGA and FGA. In formal terms one often says a scheme X over a base S is projective if there exists a quasi-coherent sheaf giving an embedding into P^n_S for some integer n, a perspective developed alongside work of Jean-Louis Koszul and Alexander Grothendieck's collaborators in the seminar series at the Institut des Hautes Études Scientifiques. The definition interacts with concepts from Noetherian ring theory, graded ring constructions, and classical projective embeddings studied by Federigo Enriques and Guido Castelnuovo.
Construction via Proj
The construction Proj of a graded ring was introduced in EGA II and further elaborated in the lectures of Grothendieck and Jean-Pierre Serre; it produces a scheme from a graded algebra A = ⊕_{d≥0} A_d. For a finitely generated graded algebra over a ring R, Proj A yields a scheme locally modelled on spectra of degree-zero parts of localizations, linking to the graded pieces studied by David Mumford in his work on geometric invariant theory and by Oscar Zariski in birational geometry. The Proj construction is inherently compatible with base change in families considered by Alexander Grothendieck and applied in moduli problems explored by Deligne and Drinfeld. Proj is used to define the relative projective space P^n_S over a scheme S, a construction exploited by Mikhail Gromov and Pierre Deligne in deformation theory contexts.
Examples
Classical examples include projective n-space P^n_k over a field k, projective hypersurfaces such as Fermat curve families and Cubic surface examples studied by Cayley and Salmon, and smooth projective curves like those in Riemann–Roch theorem contexts by Riemann and Noether. Projective toric varieties built from fans are central in work by David Cox and Gelfand and appear in mirror symmetry studies by Kontsevich and Maxim Kontsevich's collaborators. Projective schemes include classical projective varieties such as Elliptic curve models related to Weierstrass equations, projective models of K3 surfaces studied by Shioda and Piatetski-Shapiro, and Hilbert schemes like Hilbert scheme of points considered by Grothendieck and Fogarty.
Properties
Projective schemes are proper over their base, a property tied to the Valuative criterion for properness used by Grothendieck and Nagata. Ampleness of line bundles on projective schemes echoes work of Claude Chevalley and Jean-Pierre Serre on cohomological vanishing, while the notion of very ample invertible sheaf traces to embeddings studied by Castelnuovo and Mumford. The study of singularities on projective schemes connects to contributions by Hironaka on resolution of singularities and to classification programs by Shafarevich and Mori. Intersection theory on projective schemes draws on William Fulton's formalism and on foundational results by Samuel and Chevalley.
Morphisms and Functoriality
Morphisms between projective schemes respect Proj constructions and graded ring homomorphisms, a functoriality central to the work of Grothendieck in the formulation of representable functors and moduli spaces. Projective morphisms are proper and of finite type; these notions are pivotal in the study of the Hilbert scheme by Grothendieck and in the construction of Moduli spaces such as Moduli of curves by Deligne and Mumford. The concept of projective bundle arises in the context of Grothendieck's splitting principle and is used in vector bundle classification problems investigated by Atiyah and Bott.
Cohomology and Line Bundles
Cohomology of coherent sheaves on projective schemes is governed by theorems of Serre and vanishing results such as Serre vanishing; these tools were essential to Mumford's geometric invariant theory and to advances by Hartshorne in his textbook development. The Picard group of a projective scheme and the classification of ample and nef line bundles feature in the work of Fulton and Kollár and are crucial for statements like the Riemann–Roch theorem in higher dimensions proven by Hirzebruch and extended by Grothendieck. Cohomological methods on projective schemes interface with Hodge theory as developed by Griffiths and Deligne and with derived category techniques pioneered by Bondal, Orlov, and Beilinson.