Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)
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The Hessian discriminant
Full text linkWe express the Hessian discriminant of a cubic surface in terms of fundamental invariants. This answers Question 15 from the 27 questions on the cubic surface
Towards the degree of the PGL(4)-orbit of a cubic surface
Full text linkWe study the action of the group PGL(4) on the parameter space P 19of complex cubic surfaces. Specifically, we look at how the techniquesused by Aluffi and Faber in [1] can be extended to compute the degree ofthe orbit closure O of a general cubic surface. We study the base locus ofthe induced rational map P15 ⤏ O ⊂ P19 , and the first steps in resolvingthis rational map by successively blowing up the reduced base locus
On realizability of lines on tropical cubic surfaces and the Brundu-Logar normal form
Full text linkWe present results on the relative realizability of infinite families of lines on general smooth tropical cubic surfaces. Inspired by the problem of relative realizability of lines on surfaces, we investigate the information we can derive tropically from the Brundu-Logar normal form of smooth cubic surfaces. In particular, we prove that for a residue field of characteristic ≠ 2 the tropicalization of the Brundu-Logar normal form is not smooth. We also take first steps in investigating the behavior of the tropicalized lines
A tropical count of binodal cubic surfaces
Full text linkThere are 280 binodal cubic surfaces passing through 17 general points. For points in Mikhalkin position, we show that 214 of these give tropicalizations such that the nodes are separated on the tropical cubic surface
Ranks and singularities of cubic surfaces
Full text linkWe explore the connection between the rank of a polynomial and the singularities of its vanishing locus. We first describe the singularity of generic polynomials of fixed rank. We then focus on cubic surfaces. Cubic surfaces with isolated singularities are known to fall into 22 singularity types. We compute the rank of a cubic surface of each singularity type. This enables us to find the possible singular loci of a cubic surface of fixed rank. Finally, we study connections to the Hessian discriminant. We show that a cubic surface with singularities that are not ordinary double points lies on the Hessian discriminant, and that the Hessian discriminant is the closure of the rank six cubic surfaces
96120: The degree of the linear orbit of a cubic surface
Full text linkThe projective linear group PGL(C,4) acts on cubic surfaces, considered as points of PC19. We compute the degree of the 15-dimensional projective variety defined by the Zariski closure of the orbit of a general cubic surface. The result, 96120, is obtained using methods from numerical algebraic geometry
Computing invariants of cubic surfaces
Full text linkWe report on the computation of invariants, covariants, and contravariants of cubic surfaces. The approach is based on the Clebsch transfer principle and transvection. All algorithms are implemented in the computer algebra system magma. The code can be used to efficiently compute invariants of surfaces definied over number fields and function fields
Twenty-seven questions about the cubic surface
Full text linkWe present a collection of research questions on cubic surfaces in 3-space. These questions inspired the present collection of papers. This article serves as the introduction to the issue. The number of questions is meant to match the number of lines on a cubic surface. We end with a list of problems that are open
A short note on Cayley-Salmon equations
Full text linkA Cayley-Salmon equation for a smooth cubic surface S in P3 is an expression of the form l1l2l3 - m1m2m3 = 0 such that the zero set is S and li, mj are homogeneous linear forms. This expression was first used by Cayley and Salmon to study the incidence relations of the 27 lines on S. There are 120 essentially distinct Cayley-Salmon equations for S. In this note we give an exposition of a classical proof of this fact. We illustrate the explicit calculation to obtain these equations and we apply it to the Clebsch surface and to the octanomial form appearing in work of Panizzut, Sertöz and Sturmfels. Finally we show that these 120 Cayley-Salom equations can be directly computed using recent work by Cueto and Deopurkar
Maltitudes, anticenters and altitudes of cyclic polygons and polyhedra
Full text linkIn this paper we extend to cyclical polygons the concepts of maltitude and anti-center of a cyclic quadrilateral and to cyclical polyhedra the concepts of Monge plane and Monge point of a tetrahedron. In addition, we find several properties of these new concepts. The research is conducted through the study of m-point systems, starting from the results on the centroids and the medians of these systems