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Syzygies of some quadratic varieties and their connection with the cohomology of Lie algebras. (English. Russian original) Zbl 1165.14037
Russ. Math. Surv. 61, No. 5, 990-992 (2006); translation from Usp. Mat. Nauk 61, No. 5, 189-190 (2006).
Let \(i:M\hookrightarrow \mathbb{P}(V)\) be an embedding of a variety \(M\) into a projective space given by quadratic equations. The paper deals with the computation of the syzygies of this embedding. The first result is when \(U\) is a plane and \(i\) is the Veronese embedding \(\mathbb{P}(U)=\mathbb{P}^1\hookrightarrow \mathbb{P}^n=\mathbb{P}(S^nU).\) The second result is for the Plücker embedding \(i_{Gr}:Gr(2,N)\hookrightarrow\mathbb{P}(\mathop\bigwedge\limits^2W^*)\), where \(W\) is an \(N\)-dimensional vector space.
14N05 Projective techniques in algebraic geometry
13D02 Syzygies, resolutions, complexes and commutative rings
17B56 Cohomology of Lie (super)algebras
17B65 Infinite-dimensional Lie (super)algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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