Construction of Hilbert and Quot Schemes

By N. Nitsure. 2005

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This is an expository account of Grothendieck's construction of Hilbert and Quot Schemes, following his talk 'Techniques de construction et theoremes d'existence en geometrie algebriques IV : les schemas de Hilbert', Seminaire Bourbaki 221 (1960/61), together with further developments by Mumford and by Altman and Kleiman. Hilbert and Quot schemes are fundamental to modern Algebraic Geometry, in particular, for deformation theory and moduli constructions. These notes are based on a series of six lectures in the summer school 'Advanced Basic Algebraic Geometry', held at the Abdus Salam International Centre for Theoretical Physics, Trieste, in July 2003. Any scheme X defines a contravariant functor hX (called the functor of points of the scheme X) from the category of schemes to the category of sets, which associates to any scheme T the set Mor(T, X) of all morphisms from T to X. The scheme X can be recovered (up to a unique isomorphism) from hX by the Yoneda lemma. In fact, it is enough to know the restriction of this functor to the full subcategory consisting

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/6474c87549bfacb51cabe75806bf489ca72e01b4

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Citations in the corpus

• Weil's Conjecture for Function Fields (2019, background): “Let Fl(P) denote the algebraic space obtained by Weil restriction of B\P along the map XR → SpecR (for a general discussion of Weil restriction, we refer the reader to [41]).”