Given a real cubic form f(x,y,z), there is a pseudo-Riemannian metric given by its Hessian matrix, defined on the open subset of R^3 where the Hessian determinant h is non-zero. We determine the full curvature tensor of this metric in terms of h and the S-invariant of f, obtaining in the process various different characterizations of S. Motivated by the case of intersection forms associated with complete intersection threefolds in the product of three projective spaces, we then study ternary cubic forms which arise as follows: we choose positive integers d1, d2, d3, set r = d1 + d2 + d3 - 3, and consider the coefficient F(x,y,z) of H1^d1 H2^d2 H3^d3 in the product (x H1 + y H2 + z H3)^3 (a_1 H1 + b_1 H2 + c_1 H3) ... (a_r H1 + b_r H2 + c_r H3), the a_j, b_j and c_j denoting non-negative real numbers; we assume also that F is non-degenerate. Previous work of the author on sectional curvatures of Kahler moduli suggests a number of combinatorial conjectures concerning the invariants of F. It is proved here for instance that the Hessian determinant, considered as a polynomial in x,y,z and the a_j, b_j, c_j, has only positive coefficients. The same property is also conjectured to hold for the S-invariant; the evidence and background to this conjecture is explained in detail in the paper.
Cubic forms in three variables are parametrised by points of $¶^9$. We study the subvarieties in this space defined by decomposable forms. Specifically, we calculate the equivariant minimal resolutions of these varieties and describe their ideals invariant-theoretically.
In this work we give the classification of real ternary cubic forms with respect to rank and border rank up to SL(3)-action and we examine the differences with the complex case.
In this article, we provide an overview of a one-to-one correspondence between representations of the generalized Clifford algebra $C_f$ of a ternary cubic form $f$ and certain vector bundles (called Ulrich bundles) on a cubic surface $X$. We study general properties of Ulrich bundles, and using a recent classification of Casanellas and Hartshorne, deduce the existence of irreducible representations of $C_f$ of every possible dimension.
We prove that two general ternary forms are simultaneously identifiable only in the classical cases of two quadratic and a cubic and a quadratic form. We translate the problem into the study of a certain linear system on a projective bundle on the plane, and we apply techniques from projective and birational geometry to prove that the associated map is not birational.
Let ${\mathcal Q}_n^d$ be the vector space of forms of degree $d\ge 3$ on ${\mathbb C}^n$, with $n\ge 2$. The object of our study is the map $Φ$, introduced in papers [EI], [AI1], that assigns every nondegenerate form in ${\mathcal Q}_n^d$ the so-called associated form, which is an element of ${\mathcal Q}_n^{n(d-2)*}$. We focus on two cases: those of binary quartics ($n=2$, $d=4$) and ternary cubics ($n=3$, $d=3$). In these situations the map $Φ$ induces a rational equivariant involution on the projectivized space ${\mathbb P}({\mathcal Q}_n^d)$, which is in fact the only nontrivial rational …
We parametrize quartic commutative algebras over any base ring or scheme (equivalently finite, flat degree four $S$-schemes), with their cubic resolvents, by pairs of ternary quadratic forms over the base. This generalizes Bhargava's parametrization of quartic rings with their cubic resolvent rings over $\mathbb{Z}$ by pairs of integral ternary quadratic forms, as well as Casnati and Ekedahl's construction of Gorenstein quartic covers by certain rank 2 families of ternary quadratic forms. We give a geometric construction of a quartic algebra from any pair of ternary quadratic forms, and prove …
When all ternary cubic forms over $\mathbb Z$ are ordered by the heights of their coefficients, we show that a positive proportion of them fail the Hasse principle, i.e., they have a zero over every completion of $\mathbb Q$ but no zero over $\mathbb Q$. We also show that a positive proportion of all ternary cubic forms over $\mathbb Z$ nontrivially satisfy the Hasse principle, i.e., they possess a zero over every completion of $\mathbb Q$ and also possess a zero over $\mathbb Q$. Analogous results are proven for other genus one models, namely, for equations of the form $z^2=f(x,y)$ where $f$ …
We study nonnegative (psd) real sextic forms $q(x_0,x_1,x_2)$ that are not sums of squares (sos). Such a form has at most ten real zeros. We give a complete and explicit characterization of all sets $S\subset\mathbb{P}^2(\mathbb{R})$ with $|S|=9$ for which there is a psd non-sos sextic vanishing in $S$. Roughly, on every plane cubic $X$ with only real nodes there is a certain natural divisor class $τ_X$ of degree~$9$, and $S$ is the real zero set of some psd non-sos sextic if, and only if, there is a unique cubic $X$ through $S$ and $S$ represents the class $τ_X$ on $X$. If this is the case, t…
We study real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained for quadrics and cubics. For quintics we determine the real rank boundary: it is a hypersurface of degree 168. For quartics, sextics and septics we identify some of the components of the real rank boundary. The real varieties of sums of powers are stratified by discriminants that are derived from hyperdeterminants.
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.
We study typical ranks with respect to a real variety $X$. Examples of such are tensor rank ($X$ is the Segre variety) and symmetric tensor rank ($X$ is the Veronese variety). We show that any rank between the minimal typical rank and the maximal typical rank is also typical. We investigate typical ranks of $n$-variate symmetric tensors of order $d$, or equivalently homogeneous polynomials of degree $d$ in $n$ variables, for small values of $n$ and $d$. We show that $4$ is the unique typical rank of real ternary cubics, and quaternary cubics have typical ranks $5$ and $6$ only. For ternary qua…
In this paper we study the combinatorics of quasi-trigonometric solutions of the classical Yang-Baxter equation, arising from simple vector bundles on a nodal Weierstrass cubic.
Common Real Secants to Pairs of Real Twisted Cubic Curves
University of Agriculture, Faisalabad（费萨拉巴德农业大学）；Michigan State University（密歇根州立大学）；University of Notre Dame（圣母大学）；Texas State University（德州州立大学）；Grand Valley State University（大谷州立大学）；University of California, Santa Cruz（加州大学圣克鲁兹分校）
I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, i prove that t is greater or equal than 2c+1, if d is odd and t is greater or equal than max(3,2c+1), is d is even, where c is the number of ovals of the locus of zeros of f; for binary forms, i prove that t is greater or equal than the number of real roots of f. Moreover, the above inequalities are the best possible for binary forms of any degree and for ternary cubic and quartic forms.
Let $K$ be an arbitrary field of characteristic zero, $P_n:= K[ x_1, ..., x_n]$ be a polynomial algebra, and $P_{n, x_1}:= K[x_1^{-1}, x_1, ..., x_n]$, for $n\geq 2$. Let $\s' \in {\rm Aut}_K(P_n)$ be given by $$ x_1\mapsto x_1-1, \quad x_1\mapsto x_2+x_1,\quad ... ,\quad x_n\mapsto x_n+x_{n-1}.$$ It is proved that the algebra of invariants, $F_n':= P_n^{\s'}$, is a polynomial algebra in $n-1$ variables which is generated by $[\frac{n}{2}]$ quadratic and $[\frac{n-1}{2}]$ cubic (free) generators that are given explicitly. Let $\s \in {\rm Aut}_K(P_n)$ be given by %$$\s \in {\rm Aut}_K(P_n): $$…
It was conjectured in a recent article by M. Eastwood and the second author that all absolute classical invariants of forms of degree $m\ge 3$ on ${\mathbb C}^n$ can be extracted, in a canonical way, from those of forms of degree $n(m-2)$ by means of assigning every form with non-vanishing discriminant the so-called associated form. In that paper, this surprising conjecture was confirmed for binary forms of degree $m \le 6$ and ternary cubics. In the present article, we settle the conjecture in full generality. In addition, we propose a stronger version of this statement and obtain evidence su…
We give some new canonical representations for forms over $\cc$. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in $(x_1,...,x_n)$ can be written uniquely as a sum of the cubes of linear forms $\ell_{ij}(x_i,...,x_j)$, $1 \le i \le j \le n$. A general ternary quartic form is the sum of the square of a quadratic form and three fourth powers of linear forms. The methods are classical and elementary.
