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The canonical module of GT-varieties and the normal bundle of RL-varieties. (English) Zbl 1483.14084

A Galois-Togliatti system, or GT-system, is a homogeneous ideal in the polynomial ring \(K[x_0,\ldots,x_n]\), with coefficients in a field \(K\), generated by the invariant forms of degree \(d\) of a given diagonal linear action of the cyclic group of order \(d\). GT-systems were first introduced in [E. Mezzetti and R. M. Miró-Roig, J. Algebra 509, 263–291 (2018; Zbl 1395.13019)], where it was proved that they fail the Weak Lefschetz Property in degree \(d-1\). Successively in [L. Colarte-Gómez et al., Ann. Mat. Pura Appl. (4) 200, No. 4, 1757–1780 (2021; Zbl 1470.14095)] the homogeneous ideal \(I(X_d)\) of the variety \(X_d\) parameterized by a GT-system \(I_d\) was completely described in the case \(n=2\); moreover it was proved that, for any number of variables, these varieties are arithmeticaly Cohen-Macaulay.
In this article the authors prove that the ideal of any variety \(X_d\) is generated by binomials of degree \(\leq 3\); they also exhibit examples where \(I(X_d)\) is minimally generated only by quadrics, and others where it is generated by quadrics and cubics. They then describe the canonical module \(\omega_{X_d}\) of \(X_d\), proving that it is generated by monomials of degree \(d\) and \(2d\) in the relative interior of the ideal \(I_d\). As a consequence they compute the Castelnuovo-Mumford regularity of these varieties.
In the last part of the article they introduce the RL-variety \(\mathcal X_d\) associated to \(X_d\), assuming that its ideal is level. It is a smooth rational variety defined as the image of the morphism induced by the inverse system \(\mathrm{relint}(I_d)^{-1}\). As an application of a recent result of [A. Alzati and R. Re, Commun. Algebra 48, No. 6, 2492–2516 (2020; Zbl 1440.14199)], they are able to compute the cohomology of the normal bundle of RL-varieties.

MSC:

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14L30 Group actions on varieties or schemes (quotients)
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
13A50 Actions of groups on commutative rings; invariant theory
13C14 Cohen-Macaulay modules
13E15 Commutative rings and modules of finite generation or presentation; number of generators

Software:

Macaulay2
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References:

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