Covers of algebraic varieties. V: Examples of covers of degree 8 and 9 as catalecticant loci
[For part IV of this paper see \textit{G. Casnati}, Forum Math. 13, 21-36 (2001; Zbl 0971.14014)]. The author continues his analysis of coverings of low degree of algebraic varieties. He approaches the problem ``from the bottom'', namely, given a variety \(Y\), he tries to encode finite flat coverings \(f\colon X\to Y\) of fixed degree via a set of geometrical ``data'' (i.e., vector bundles, global sections,\dots) on \(Y\). This problem has a complete answer, at least for Gorenstein covers, if the degree of the cover is at most 5 [cf. \textit{G. Casnati} and \textit{T. Ekedahl}, J. Algebr. Geom. 5, 439-460 (1996; Zbl 0866.14009), \textit{G. Casnati}, J. Algebr. Geom. 5, 461-477 (1996; Zbl 0921.14006)]. In part IV of the paper (loc. cit.), the author has given a construction of a class of degree 6 covers. Here he describes constructions giving covers of degree 8 and 9. The common feature of these examples and of those described in the previous papers is the following: Let \(d\) be the degree of the cover; there are a \({\mathbb{P}}^{n+d-2}\)-bundle \(\tilde{\mathbb{P}}\) on \(Y\), a subbundle \({\mathbb{P}}\subset \tilde{\mathbb{P}}\) of rank \(d-2\) and a variety \(V\subset {\mathbb{P}}\) such that: (a) \(V\) has relative dimension \(n\) and degree \(d\); (b) for every \(y\in Y\) the fibre \(V_y\) of \(V\) over \(y\) is a Del Pezzo \(n\)-fold; (c) \({\mathbb{P}}\cap V_y\) has dimension 0 for every \(y\in Y\). Then one sets \(X:=V\cap {\mathbb{P}}\) and the natural map \(X\to Y\) is a cover of degree \(d\). In the case of degree 8, for fixed \(y\in Y\) one can identify \(\tilde{\mathbb{P}}_y\) with the projectivization of the space of symmetric matrices of order 4 and \(V_y\) with the set of matrices of rank 1. In the case of degree 9, \(\tilde{\mathbb{P}}_y\) is the projectivized space of trilinear symmetric forms on a \(3\)-dimensional vector space and \(V_y\) is a suitably defined degeneracy locus for these maps. The paper contains also some Bertini type statements, showing that, given a curve or a surface \(Y\), the proposed constructions of covers of degree 8 and 9 actually give smooth covers of \(Y\).
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