86 research outputs found
On higher Gauss maps
We prove that the general fibre of the -th Gauss map has dimension if
and only if at the general point the -th fundamental form consists of
cones with vertex a fixed , extending a known theorem for the
usual Gauss map. We prove this via a recursive formula for expressing higher
fundamental forms. We also show some consequences of these results.Comment: 12 pages, AMS-LaTeX; to appear in the Journal of Pure and Applied
Algebr
Togliatti systems
We find some examples in P 5 (C) of surfaces satisfying Laplace equations. In particular, we study rational surfaces in P 5 (C) whose hyperplane sections have genus one that satisfy a Laplace equation. Then we study monomial Togliatti systems of cubics for variety of dimension three, i.e. we find all the monomial examples of three-folds satisfying Laplace equations
On the Hilbert vector of the Jacobian module of a plane curve
We identify several classes of curves , for which the Hilbert vector
of the Jacobian module can be completely determined, namely the 3-syzygy
curves, the maximal Tjurina curves and the nodal curves, having only rational
irreducible components. A result due to Hartshorne, on the cohomology of some
rank 2 vector bundles on , is used to get a sharp lower bound for
the initial degree of the Jacobian module , under a semistability
condition.Comment: 10 pages, 4 figures. To appear in Portugaliae Mathematic
Geometry of syzygies via Poncelet varieties
We consider the Grassmannian of -dimensional linear
subspaces of V_n=H^0({\P^1},\O_{\P^1}(n)). We define as
the classifying space of the -dimensional linear systems of degree on
whose basis realize a fixed number of polynomial relations of fixed
degree, say a fixed number of syzygies of a certain degree. The first result of
this paper is the computation of the dimension of . In the
second part we make a link between and the Poncelet
varieties. In particular, we prove that the existence of linear syzygies
implies the existence of singularities on the Poncelet varieties
Singular hypersurfaces characterizing the Lefschetz properties
In the paper untitled "Laplace equations and the Weak Lefschetz Property" the
authors highlight the link between rational varieties satisfying a Laplace
equation and artinian ideals that fail the Weak Lefschetz property. Continuing
their work we extend this link to the more general situation of artinian ideals
failing the Strong Lefschetz Property. We characterize the failure of SLP (that
includes WLP) by the existence of special singular hypersurfaces (cones for
WLP). This characterization allows us to solve three problems posed by Migliore
and Nagel and to give new examples of ideals failing the SLP. Finally, line
arrangements are related to artinian ideals and the unstability of the
associated derivation bundle is linked with the failure of SLP. Moreover we
reformulate the so-called Terao's conjecture for free line arrangements in
terms of artinian ideals failing the SLP
Lefschetz Properties for Higher Order Nagata Idealizations
We study a generalization of Nagata idealization for level algebras. These
algebras are standard graded Artinian algebras whose Macaulay dual generator is
given explicity as a bigraded polynomial of bidegree . We consider the
algebra associated to polynomials of the same type of bidegree . We
prove that the geometry of the Nagata hypersurface of order is very similar
to the geometry of the original hypersurface. We study the Lefschetz properties
for Nagata idealizations of order , proving that WLP holds if .
We give a complete description of the associated algebra in the monomial square
free case.Comment: 16 pages, 4 figures. To appear in Advances in Applied Mathematic
Newton's lemma for differential equations
The Newton method for plane algebraic curves is based on the following remark: the first term of a series, root of a polynomial with coefficients in the ring of series in one variable, is a solution of an initial equation that can be determined by the Newton polygon. Given a monomial ordering in the ring of polynomials in several variables, we describe the systems of initial equations that satisfy the first terms of the solutions of a system of partial differential equations. As a consequence, we extend Mora and Robbiano’s Groebner fan to differential ideals
On the duals of smooth projective complex hypersurfaces
We show in this note that a generic hypersurface of degree in
the complex projective space of dimension has at
least one hyperplane section containing exactly ordinary double
points, alias singularities, in general position, and no other
singularities. Equivalently, the dual hypersurface has at least one
normal crossing singularity of multiplicity . Using this result, we show
that the dual of any smooth hypersurface with has at least a very
singular point , in particular a point of multiplicity .Comment: v2. Theorem 1.4 is new and says something about the dual of any
smooth hypersurface. Hence the slight change in the name of the paper. arXiv
admin note: substantial text overlap with arXiv:2202.0223
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