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Fano congruences of index 3 and alternating 3-forms. (Congruences de droites de Fano d’indice 3 et 3-formes alternées.) (English. French summary) Zbl 1407.14046

Consider a vector space \(V\) of dimension \(n+1\) and the corresponding projective space \(\mathbb P^n=\mathbb P(V)\). A congruence of lines in \(\mathbb P(V)\) is an algebraic family of lines of dimension \(n-1\), which can be identified with an \((n-1)\)-dimensional subvariety of the Grassmannian \(\mathrm{Gr}(2,V)\). A general alternating \(3\)-form \(\omega\in\bigwedge^3V^*\) determines a linear subspace \(L_\omega\) of codimension \((n+1)\) in the span \(\mathbb P(\bigwedge^2V)\) of \(\mathrm{Gr}(2,V)\) which meets \(\mathrm{Gr}(2,V)\) in a congruence of lines \(X_\omega\). These subvarieties \(X_\omega\), for \(\omega\) general, are Fano varieties of index \(3\), and their classification was known (at least partially) only up to \(n=9\). The authors study the geometric properties of congruences \(X_\omega\), detemined by general alternating \(3\)-forms, for any \(n\). \(X_\omega\) turns out to be smooth, subcanonical and arithmetically Cohen-Macaulay, hence arithmetically Gorenstein. Moreover, the authors determine invariants and equations of the fundamental locus of \(X_\omega\), defined as the set of points in \(\mathbb P(V)\) contained in infinitely many lines of the congruence. For a general choice of \(x,y\in V^*\), the linear space \(\Lambda_{\omega,x\wedge y}=\{L\in \bigwedge^2V: \omega(L)\wedge x \wedge y =0\}\) contains \(X_\omega\). The residual \(Y_{\omega,x,y}\) of \(X_\omega\) in the intersection \(\mathrm{Gr}(2,V)\cap \Lambda_{\omega,x\wedge y}\) is another (singular) congruence of lines, and the authors determine several properties of \(Y_{\omega,x\wedge y}\) too.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14J45 Fano varieties
14M06 Linkage

Keywords:

congruences
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References:

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