Gaps in the number of generators of monomial Togliatti systems

Almeida, Charles; Andrade, Ana Paula Rocha Carvalho Bernardes de; Miró‐Roig, Rosa M.

doi:10.1016/j.jpaa.2018.07.009

Cited by 9 publications

(18 citation statements)

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“…The name is in honour of Togliatti who proved that, for n = 2, the only smooth Togliatti system of cubics is [28,29]). The systematic study of Togliatti systems was initiated in [20] and it has been continuing in [1,18,19,21,22]. Precisely, in [18], it was introduced the notion of GT-system with group a finite cyclic group.…”

Section: Gt-systems and Gt-varietiesmentioning

confidence: 99%

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The Canonical Module of GT-Varieties and the Normal Bundle of RL-Varieties

Colarte-Gómez

Miró‐Roig

2022

Mediterr. J. Math.

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In this paper, we study the geometry of GT-varieties $$X_{d}$$ X d with group a finite cyclic group $$\Gamma \subset {{\,\mathrm{GL}\,}}(n+1,\mathbb {K})$$ Γ ⊂ GL ( n + 1 , K ) of order d. We prove that the homogeneous ideal $${{\,\mathrm{I}\,}}(X_{d})$$ I ( X d ) of $$X_{d}$$ X d is generated by binomials of degree at most 3 and we provide examples reaching this bound. We give a combinatorial description of the canonical module of the homogeneous coordinate ring of $$X_{d}$$ X d and we show that it is generated by monomial invariants of $$\Gamma $$ Γ of degree d and 2d. This allows us to characterize the Castelnuovo–Mumford regularity of the homogeneous coordinate ring of $$X_d$$ X d . Finally, we compute the cohomology table of the normal bundle of the so-called RL-varieties. They are projections of the Veronese variety $$\nu _{d}(\mathbb {P}^{n}) \subset \mathbb {P}^{\left( {\begin{array}{c}n+d\\ n\end{array}}\right) -1}$$ ν d ( P n ) ⊂ P n + d n - 1 which naturally arise from level GT-varieties.

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Section: Gt-systems and Gt-varietiesmentioning

confidence: 99%

“…They called I a Togliatti system. Since then, Togliatti systems have been extensively studied as one can see in [1,6,9,18,19,21,22].…”

Section: Introductionmentioning

confidence: 99%

The Canonical Module of GT-Varieties and the Normal Bundle of RL-Varieties

Colarte-Gómez

Miró‐Roig

2022

Mediterr. J. Math.

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“…(see [3], [26] and [27]). The systematic study of Togliatti systems was initiated in [15] and for recent results the reader can see [18], [16], [1], [20] and [17]. Precisely in the latter reference the authors introduced the notion of GT-system which we recall now.…”

Section: On the Minimality Of Gt-systemsmentioning

confidence: 99%

“…Our main result is Theorem 2.3, where we prove that conditions (1) are sufficient in the case of 3-lines circulant matrices of order d, of the form Circ(x, 0, . .…”

Section: Introductionmentioning

confidence: 96%