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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.
##### MSC:
 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
##### Keywords:
Syzygies; cohomology of Lie algebras
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