Combinatorial Algebraic Geometry: Levico Terme, Italy 2013, by Aldo Conca

By Aldo Conca

Combinatorics and Algebraic Geometry have loved a fruitful interaction because the 19th century. Classical interactions contain invariant concept, theta features and enumerative geometry. the purpose of this quantity is to introduce fresh advancements in combinatorial algebraic geometry and to procedure algebraic geometry with a view in the direction of purposes, resembling tensor calculus and algebraic records. a standard subject is the research of algebraic kinds endowed with a wealthy combinatorial constitution. suitable suggestions comprise polyhedral geometry, unfastened resolutions, multilinear algebra, projective duality and compactifications.

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Extra info for Combinatorial Algebraic Geometry: Levico Terme, Italy 2013, Editors: Sandra Di Rocco, Bernd Sturmfels

Example text

Thus let K be a field and let R1 ; R2 ; : : : be commutative K-algebras with 1. The algebra Ri plays the role of coordinate ring of the ambient space of the i -th variety in our chain. Assume that the Ri are linked by (unital) ring homomorphisms Ãi W Ri ! Ri C1 and iSW Ri C1 ! Ri satisfying i ı Ãi D 1Ri . Then we can form the K-algebra R1 WD i 2N Ri with respect to the inclusions Ãi ; the use of the i will become clear later. Suppose, next, that are given ideals Ii  Ri such that i maps Ii C1 into Ii and Ãi maps Ii into Ii C1 .

This turns out to be a difficult question on its own. 18. 3; 3; 2/ is Koszul. The same holds for the generic projection of the Veronese surface of P9 to P8 . 3; 3; 2/ is G-quadratic. 2]. n; n; n 1/ are shellable for n > 3, see [Ta]. It is not clear whether the same is true for n D 3. 3 Koszul Algebras Associated with Hyperspace Configurations Another interesting family of Koszul algebras with relations to combinatorics arises in the following way. Let V D V1 ; : : : ; Vm be a collection of subspaces of the space of linear forms in the polynomial ring KŒx1 ; : : : ; xn .

Let … be a submonoid of G and let Z be a …-stable subset of X S. Assume that Z is …-Noetherian with the induced topology. Then Y WD ZG D g2G Zg  X is G-Noetherian with the induced topology. Proof. Let Y D Y1 à Y2 à Y3 à : : : be a chain of G-stable closed subsets of Y . Then each Zi WD Yi \ Z is …-stable and closed, hence by …-Noetherianity of Z there exists an n with Zn D ZnC1 D : : :. By definition of Y , for each y 2 Yi there exist a g 2 G and a z 2 Z with y D zg, and by G-stability of Yi we have z D yg 1 2 Zi .

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