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Betti strata of height two ideals. (English) Zbl 1095.13014
Let \(R\) be the polynomial ring in two variables over an infinite field \(k\), and \(H\) the Hilbert function of some graded quotients of \(R\). The variety \(G(H)\) parametrizes graded ideals \(I\) in \(R\) of Hilbert function \(H(R/I,-)=H\). It is a closed subvariety of a finite product of Grassmannians. The Betti stratum \(G_{\beta}(H)\) parametrizes all graded quotients \(R/I\) of \(R\) having fixed \(\beta_i=\dim_k\text{Tor}_1^R(I,k)_i\). The Betti strata are irreducible and Cod\((G_{\beta}(H))=\Sigma_{i>\mu}\beta_i\nu_i\), where \(\mu=\min\{i:H_i<i+1\}\) and \(\nu_i\) is the number of generators of degree \(i\) in \(I\) (\(H\) and \(\beta\) determine uniquely \(\nu\)). When \(k\) is algebraically closed the closure of a Betti stratum is the union of more special strata and is Cohen-Macaulay. Given the socle type, all possible Hilbert functions of Artinian graded quotients of \(R\) are determined.

MSC:
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14M15 Grassmannians, Schubert varieties, flag manifolds
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D02 Syzygies, resolutions, complexes and commutative rings
14C05 Parametrization (Chow and Hilbert schemes)
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