Abstract configurations in algebraic geometry. (English) Zbl 1068.14059

Dolgachev, Igor V.

Collino, Alberto (ed.) et al., The Fano conference. Papers of the conference organized to commemorate the 50th anniversary of the death of Gino Fano (1871–1952), Torino, Italy, September 29–October 5, 2002. Torino: Università di Torino, Dipartimento di Matematica. 423-462 (2004).

An abstract configuration is a combinatorial object whose investigation is a fundamental topic for many applications of finite geometries. It consists of a correspondence \(R\) between two finite sets \(A,B\), such that the cardinality of \(R(a)\) and \(R^{-1}(b)\) does not depend on the choice of \(a\in A\), \(b\in B\). If in addition the sets \(R(a)\cap R(a')\) are non-empty, of fixed cardinality, then the configuration is a block-design, an object with interesting applications in many fields.
The natural geometrical interpretation of \(R\) is by means of an incidence correspondence. The author defines a projective interpretation of \(R\) as a projective variety \(X\) such that \(R\) can be read as the incidence correspondence between two sets of subvarieties \(A,B\) of fixed dimension \(d_A,d_B\). Then the author introduces and describes several old and new projective interpretations, relevant both for their combinatorial and geometrical background. Among them are the classical examples of the Kummer configuration of \(16\) points and tropes and the Hasse configuration based on the inflection points of a plane cubic.
For the entire collection see [Zbl 1051.00013].

Reviewer: Luca Chiantini (Siena)

MSC:

14N20	Configurations and arrangements of linear subspaces
14G15	Finite ground fields in algebraic geometry