Authors:Turku Ozlum Celik; Francesco Galuppi, Avinash Kulkarni, Miruna-Ştefana Sorea Abstract: We show that the eigenschemes of 4 × 4 × 4 symmetric tensors are parameterized by a linear subvariety of the Grassmannian Gr(3, P14). We also study the decomposition of the eigenscheme into the subscheme associated to the zero eigenvalue and its residue. In particular, we describe the possible degrees and dimensions. PubDate: Wed, 09 Sep 2020 16:42:32 +000
Authors:Rodica Dinu; Tim Seyannaeve Abstract: We 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. PubDate: Wed, 09 Sep 2020 15:50:57 +000
Authors:Anna Seigal; Eunice Sukarto Abstract: We 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. PubDate: Wed, 09 Sep 2020 15:49:52 +000
Authors:Marvin Anas Hahn; Sara Lamboglia, Alejandro Vargas Abstract: A 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. PubDate: Wed, 09 Sep 2020 15:48:42 +000
Authors:Maria Donten-Bury; Paul Görlach, Milena Wrobel Abstract: We investigate the class of degenerations of smooth cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. More precisely, we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and who leave the question of classifying these degenerations in the degree 3 case as an open problem. In order to carry out this classification we describe an approach which is closely related to tropical geometry and present partial results in this direction. PubDate: Wed, 09 Sep 2020 15:47:27 +000
Authors:Marta Panizzut; Emre Sertöz, Bernd Sturmfels Abstract: We present a new normal form for cubic surfaces that is well suited for p-adic geometry, as it reveals the intrinsic del Pezzo combinatorics of the 27 trees in the tropicalization. The new normal form is a polynomial with eight terms, written in moduli from the E6 hyperplane arrangement. If such a surface is tropically smooth then its 27 tropical lines are distinct. We focus on explicit computations, both symbolic and p-adic numerical. PubDate: Wed, 09 Sep 2020 15:45:15 +000
Authors:Hanieh Keneshlou Abstract: The Eckardt hypersurface in P19 is the closure of the locus of smooth cubic surfaces with an Eckardt point, which is a point common to three of the 27 lines on a smooth cubic surface. We describe the cubic surfaces lying on the singular locus of the model of this hypersurface in P4, obtained via restriction to the space of cubic surfaces possessing a so-called Sylvester form. We prove that, inside the moduli of cubics, the singular locus corresponds to a reducible surface with two rational irreducible components intersecting along two rational curves. The two curves intersect at two points representing the Clebsch and the Fermat cubic surfaces. We observe that the cubic surfaces parameterized by the two components or the two rational curves are distinguished by the number of Eckardt points and automorphism groups. PubDate: Wed, 09 Sep 2020 15:43:10 +000
Authors:Dominic Bunnett; Hanieh Keneshlou Abstract: We compute and study two determinantal representations of the discriminant of a cubic quaternary form. The first representation is the Chow form of the 2-uple embedding of P3 and is computed as the Pfaffian of the Chow form of a rank 2 Ulrich bundle on this Veronese variety. We then consider the determinantal representation described by Nanson. Weinvestigate the geometric nature of cubic surfaces whose discriminant matrices satisfy certain rank conditions. As a special case of interest, we use certain minors of this matrix to suggest equations vanishing on the locus of $k$-nodal cubic surfaces. PubDate: Wed, 09 Sep 2020 15:41:59 +000
Authors:Lars Kastner; Robert Löwe Abstract: We determine the 166104 extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with D-equivalence classes of regular triangulations of the 3-dilated tetrahedron. We describe how to compute these triangulations and their D-equivalence classes in order to arrive at our main result. The computation poses several challenges, such as dealing with the sheer number of triangulations effectively, as well as devising a suitably fast algorithm for computation of a D-equivalence class. PubDate: Wed, 09 Sep 2020 15:39:50 +000
Authors:Andreas-Stephan Elsenhans; Jörg Jahnel Abstract: We 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. PubDate: Wed, 09 Sep 2020 15:26:45 +000
Authors:Elisa Cazzador; Bjørn Skauli Abstract: We study the action of the group PGL(4) on the parameter space P 19 of complex cubic surfaces. Specifically, we look at how the techniques used by Aluffi and Faber in [1] can be extended to compute the degree of the orbit closure O of a general cubic surface. We study the base locus of the induced rational map P15 ⤏ O ⊂ P19 , and the first steps in resolving this rational map by successively blowing up the reduced base locus. PubDate: Wed, 09 Sep 2020 15:23:35 +000
Authors:Laura Brustenga I Moncusì; Sascha Timme, Madeleine Weinstein Abstract: The 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. PubDate: Wed, 09 Sep 2020 15:13:52 +000
Authors:Kristian Ranestad; Bernd Sturmfels Abstract: We 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. PubDate: Wed, 09 Sep 2020 15:06:17 +000
Authors:Alheydis Geiger; Madeline Brandt Abstract: There 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. PubDate: Wed, 09 Sep 2020 00:00:00 +000
Authors:Alheydis Geiger Abstract: We 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. PubDate: Wed, 09 Sep 2020 00:00:00 +000
Open Access journal
[2 journals]
