Algebraic geometry I. From algebraic varieties to schemes

Algebraic variety
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Various definitions of geometric vector bundle. Ample sheaves. Some properties of ample sheaves, the ample cone, and qco sheaves on X when X has an ample sheaf.

scheme theory

Examples described without proof : flag variety, curves. Base change and composition of morphisms that have an ample sheaf. Basic properties of quasi-projective morphisms. Very ample bundles.

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More information about this seller Contact this seller 4. You can participate in the seminar, or in the lecture including the exercise sessions , or in both. The algebraic geometers of the Italian school had often used the somewhat foggy concept of the generic point of an algebraic variety. Primary decomposition of ideals. One of the advantages of algebraic geometry is that it is purely algebraically defined and applied to any field, including fields of finite characteristic.

Proposition: if T is a quasi-compact and quasi-separated scheme, then every qco sheaf on T is the direct limit of its locally finitely generated subsheaves. Example of elimination theory, to motivate projective and proper morphisms. More motivating examples for proper and projective morphisms. Nakayama's Lemma. Finite morphisms are closed. Universally closed and proper morphisms. Morphism yoga. Every Z -morphism from a scheme proper over Z to a scheme separated over Z is proper. Projective morphisms.

Properties of the category of schemes projective over a nice e. Classical Grassmann varieties G n,r. Start on representability of the Grassmann functor. Grassmann schemes. Top exterior power of the tautological quotient bundle is very ample for a closed projective embedding. Partial flag varieties for GL n as iterated Grassmann schemes. Classical and combinatorial-topological definitions of dimension.

Preview of dimension theory for Noetherian local rings. Every non-empty closed subset of a locallly Noetherian scheme has a closed point. Fiber dimension inequality for morphisms of locally Noetherian schemes. Behavior of dimension under a finite morphism. Example of a nice Noetherian domain localization of a polynomial ring with closed points of differing codimensions. Brief proof sketch; geometric counterexample with A not integrally closed. Fiber dimension identity for flat morphisms. Applications of dimension theory for algebraic schemes: algebraic schemes are catenary; example of using fiber dimensions to show that an algebraic scheme is irreducible.

General remarks on cohomology theories de Rham, simplicial, sheaf. Abelian categories e. Mapping cones. Long exact sequence of a mapping cone triangle.

Deligne's definition of derived functor. Computing RF A using F -acyclic resolutions. Cech complexes, objects acyclic with respect to a base of the topology, Cech cohomology. Theorem: Qco sheaves on an affine scheme are acyclic for the global sections functor.

Unions and intersections.

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Varieties in k 1 and k 2. Algebraic groups SL n and GL n. Elliptic curves and their degenerations.

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Consequences of Nullstellensatz. Morphisms of affine varieties. Correspondence between morphisms and k -algebra homomorphisms. Simple examples. A large part of my work in this area is in collaboration with Pandharipande. On the other hand, I am interested in describing all the cohomology of the moduli spaces of curves and abelian varieties.

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  7. Algebraic Geometry I References.

My area of research is commutative algebra. I have mainly been concerned with the theory of noetherian rings and modules and made some research on chain conditions, lengths, generating sets of ideals etc. I am working on various aspects of supersymmetry and their applications. Some of them are unexpected, e. I intend to describe the Lie superalgebras of classical equations of mathematical physics. I intend to describe highest weight representations of distinguished simple Lie superalgebras of string theories extending current results of B.

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Feigin et al. The answer might be interpreted in terms of critical phenomena of such materials as graphene. The new examples of simple finite dimensional Lie super algebras over fields of characteristic 2 are being obtained. I am an algebraic geometer broadly interested in questions related to moduli problems. In loose terms, a moduli space is a geometric object that parameterizes other geometric objects curves, surfaces, cycles, etc. The last few years I have among other things studied foundational questions for algebraic stacks and algebraic spaces the modern setting for moduli problems and this is an on-going interest.

My present research activities also include log geometry, root stacks, non-archimedean geometry, derived algebraic geometry, wild ramification and resolution of singularities. My research interests are in algebraic geometry and commutative algebra, more specifically Hilbert schemes, Quot schemes and related moduli problems.

Moduli spaces are algebraic objects solving particular moduli problems. The Hilbert scheme parameterizing subvarieties in a given ambient variety is an example of such a moduli space. Important for many branches in algebraic geometry is that many moduli spaces exist quite generally. My research is centered around moduli spaces, their existence and their explicit construction. Recently, together with several colleagues, I work on the construction of Quot schemes on specific varieties, on specific properties of the Quot space in general, and on the moduli of projective curves with a specific resolution.

Algebra and geometry. Sandra Di Rocco My recent research is about projective toric varieties and computational Algebraic Geometry. Carel Faber My research is in algebraic geometry; the moduli spaces of curves and their intersection theory form the main subject. Christian Gottlieb My area of research is commutative algebra.

Algebraic Geometry