Abstract. We prove that r independent homogeneous polynomials of the same degree d become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety whose (d − 1)-osculating spaces have dimension smaller than expected. This gives an equivalence between an algebraic notion (called Weak Lefschetz Property) and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case, some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture.
We study the homogeneous artinian ideals of the polynomial ring K[x, y, z] generated by the homogenous polynomials of degree d which are invariant under an action of the cyclic group Z/dZ, for any d ≥ 3. We prove that they are all monomial Togliatti systems, and that they are minimal if the action is defined by a diagonal matrix having on the diagonal (1, e, e a ), where e is a primitive d-th root of the unity. We get a complete description when d is prime or a power of a prime. We also establish the relation of these systems with linear Ceva configurations.Acknowledgments: The first author is member of INdAM -GNSAGA and is supported by PRIN "Geometria delle varietà algebriche" and by FRA, Fondi di Ricerca di Ateneo, Università di Trieste. The second author was partially supported by MTM2013-45075-P.
Abstract. We compute the minimal and the maximal bound on the number of generators of a minimal smooth monomial Togliatti system of forms of degree d in n + 1 variables, for any d ≥ 2 and n ≥ 2. We classify the Togliatti systems with number of generators reaching the lower bound or close to the lower bound. We then prove that if n = 2 (resp n = 2, 3) all range between the lower and upper bound is covered, while if n ≥ 3 (resp. n ≥ 4) there are gaps if we only consider smooth minimal Togliatti systems (resp. if we avoid the smoothness hypothesis). We finally analyze for n = 2 the Mumford-Takemoto stability of the syzygy bundle associated to smooth monomial Togliatti systems.
We study congruences of lines Xω defined by a sufficiently general choice of an alternating 3-form ω in n+1 dimensions, as Fano manifolds of index 3 and dimension n−1. These congruences include the G2-variety for n=6 and the variety of reductions of projected ℙ2×ℙ2 for n=7.\ud We compute the degree of Xω as the n-th Fine number and study the Hilbert scheme of these congruences proving that the choice of ω bijectively corresponds to Xω except when n=5. The fundamental locus of the congruence is also studied together with its singular locus: these varieties include the Coble cubic for n=8 and the Peskine variety for n=9.\ud The residual congruence Y of Xω with respect to a general linear congruence containing Xω is analysed in terms of the quadrics containing the linear span of Xω. We prove that Y is Cohen-Macaulay but non-Gorenstein in codimension 4. We also examine the fundamental locus G of Y of which we determine the singularities and the irreducible components
Given any diagonal cyclic subgroup Λ ⊂ GL(n + 1, k) of order d, let I d ⊂ k[x0, . . . , xn] be the ideal generated by all monomials {m1, . . . , mr} of degree d which are invariants of Λ. I d is a monomial Togliatti system, provided r ≤ d+n−1 n−1, and in this case the projective toric variety X d parameterized by (m1, . . . , mr) is called a GT -variety with group Λ. We prove that all these GT -varieties are arithmetically Cohen-Macaulay and we give a combinatorial expression of their Hilbert functions. In the case n = 2, we compute explicitly the Hilbert function, polynomial and series of X d . We determine a minimal free resolution of its homogeneous ideal and we show that it is a binomial prime ideal generated by quadrics and cubics. We also provide the exact number of both types of generators. Finally, we pose the problem of determining whether a surface parameterized by a Togliatti system is aCM. We construct examples that are aCM and examples that are not. Contents 1. Introduction. 2. Preliminaries. 2.1. Semigroup rings and rings of invariants 2.2. Galois coverings and quotient varieties 2.3. Lefschetz properties and Togliatti systems 3. The arithmetic Cohen-Macaulayness of GT-varieties. 4. Hilbert function of GT-surfaces. 5. A new family of aCM surfaces parameterized by monomial Togliatti systems References
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