arXiv subjects
Publications and source records attributed to Kristian Ranestad.
A general algebraic surface cannot be embedded in P4. Proving a conjecture by Hartshorne and Lichtenbaum, Ellingsrud and Peskine showed that there is a degree bound for smooth rational surfaces in P4, and in fact for surfaces not of general type. We give a survey of the classification status and of classical and computer-aided constructions of smooth non-general type surfaces in P4.
This paper studies the algebraic degree of generalized Nash equilibrium problems (GNEPs) given by polynomials. Their generalized Nash equilibria (GNEs), as well as their KKT or Fritz-John points, are algebraic functions in the coefficients of defining polynomials. We study the degrees of these algebraic functions, which also count the numbers of complex KKT or Fritz-John points. Under some genericity assumptions, we show that a GNEP has only finitely many complex Fritz-John points and every Fritz-John point is a KKT point. We also give formulae for algebraic degrees of GNEPs, which count the numbers of complex Fritz-John points for generic cases.
The classical trisecant lemma says that a general chord of a non-degenerate space curve is not a trisecant; that is, the chord only meets the curve in two points. The generalized trisecant lemma extends the result to higher-dimensional varieties. It states that the linear space spanned by general points on a projective variety intersects the variety in exactly these points, provided the dimension of the linear space is smaller than the codimension of the variety and that the variety is irreducible, reduced, and non-degenerate. We prove a real analogue of the generalized trisecant lemma, which takes the form of a trichotomy. Along the way, we characterize the possible numbers of real intersection points between a real projective variety and a complimentary dimension real linear space. We show that any integer of correct parity between a minimum and a maximum number can be achieved. We then specialize to Segre-Veronese varieties, where our results apply to the identifiability of independent component analysis, tensor decomposition and to typical tensor ranks.
In this article we study adjoint hypersurfaces of geometric objects obtained by intersecting simple polytopes with few facets in $\mathbb{P}^5$ with the Grassmannian $\mathrm{Gr}(2,4)$. These generalize the positive Grassmannian, which is the intersection of $\mathrm{Gr}(2,4)$ with the simplex. We show that if the resulting object has five facets, it is a positive geometry and the adjoint hypersurface is unique. For the case of six facets we show that the adjoint hypersurface is not necessarily unique and give an upper bound on the dimension of the family of adjoints. We illustrate our results with a range of examples. In particular, we show that even if the adjoint is not unique, a positive hexahedron can still be a positive geometry.
We show that the parameters of a $k$-mixture of inverse Gaussian or gamma distributions are algebraically identifiable from the first $3k-1$ moments, and rationally identifiable from the first $3k+2$ moments. Our proofs are based on Terracini's classification of defective surfaces, careful analysis of the intersection theory of moment varieties, and a recent result on sufficient conditions for rational identifiability of secant varieties by Massarenti--Mella.
This article serves as an introduction to the special volume on Positive Geometry in the journal Le Matematiche. We attempt to answer the question in the title by describing the origins and objects of positive geometry at this early stage of its development. We discuss the problems addressed in the volume and report on the progress. We also list some open challenges.
We study the conditions under which a subvariety of the Grassmannian may be recovered from certain of its linear projections. In the special case that our Grassmannian is projective space, this is equivalent to asking when a variety can be recovered from its Chow form; the answer is "always" by work of Chow in 1937. In the general Grassmannian setting, the analogous question is when a variety can be recovered from its Chow-Lam form. We give both necessary conditions for recovery and families of examples where, in contrast with the projective case, recovery is not possible.
Polypols are natural generalizations of polytopes, with boundaries given by nonlinear algebraic hypersurfaces. We describe polypols in the plane and in 3-space that admit a unique adjoint hypersurface and study them from an algebro-geometric perspective. We relate planar polypols to positive geometries introduced originally in particle physics, and identify the adjoint curve of a planar polypol with the numerator of the canonical differential form associated with the positive geometry. We settle several cases of a conjecture by Wachspress claiming that the adjoint curve of a regular planar polypol does not intersect its interior. In particular, we provide a complete characterization of the real topology of the adjoint curve for arbitrary convex polygons. Finally, we determine all types of planar polypols such that the rational map sending a polypol to its adjoint is finite, and explore connections of our topic with algebraic statistics.
We study the variety $\rm{VAPS}_G(q_2^3, 10)$ (resp. $\rm{VAPS}_G(q_3^2, 10)$), a Grassmannian compactification of the variety of finite schemes of length $10$ apolar to $q_{2}^3$ (resp. $q_3^2$), where $\{q_{n} = 0\}\subset \mathbb P^{n} $ is a smooth quadric hypersurface. In particular, we show that $\rm{VAPS_G}(q_2^3, 10)$ is the tangent developable of a rational normal curve, while $\rm{VAPS}_G(q_3^2, 10)$ is reducible with three singular $5$-dimensional components.
We study the curvature of a smooth algebraic surface $X\subset \mathbb R^3$ of degree $d$ from the point of view of algebraic geometry. More precisely, we consider umbilical points and points of critical curvature. We prove that the number of complex critical curvature points is of order $d^3$. For general quadrics, we fully characterize the number of real and complex umbilics and critical curvature points.
We prove that the smallest degree of an apolar 0-dimensional scheme of a general cubic form in $n+1$ variables is at most $2n+2$, when $n\geq 8$, and therefore smaller than the rank of the form. For the general reducible cubic form the smallest degree of an apolar subscheme is $n+2$, while the rank is at least $2n$.
We discuss the space $VPS(Q,H)$ of ideals with Hilbert function $H=(1,n,n, \ldots)$ that are apolar to a full rank quadric $Q$. We prove that its components of saturated ideals are closely related to the locus of Gorenstein algebras and to the Slip component in border apolarity. We also point out an important error in~[RS] and provide the necessary corrections.
We consider semi-algebraic subsets of the Grassmannian of lines in three-space called tree amplituhedra. These arise in the study of scattering amplitudes from particle physics. Our main result states that tree amplituhedra in ${\rm Gr}(2,4)$ are positive geometries. The numerator of their canonical form plays the role of the adjoint in Wachspress geometry, and is uniquely determined by explicit interpolation conditions.
This is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces do. Our method is a detailed analysis of a general purpose formula by Ranestad and Sturmfels in the case of smooth real algebraic surfaces of low degree (that are rational over the complex numbers). We study both the complex and the real features of the algebraic boundary of Veronese, Del Pezzo and Bordiga surfaces.
A quaternary quartic form, a quartic form in four variables, is the dual socle generator of an Artinian Gorenstein ring of codimension and regularity 4. We present a classification of quartic forms in terms of rank and powersum decompositions which corresponds to the classification by the Betti tables of the corresponding Artinian Gorenstein rings. This gives a stratification of the space of quaternary quartic forms which we compare with the Noether-Lefschetz stratification. We discuss various phenomena related to this stratification. We study the geometry of powersum varieties for a general form in each stratum. In particular, we show that the powersum variety $VSP(F,9)$ of a general quartic with singular middle catalecticant is again a quartic surface, thus giving a rational map between two divisors in the space of quartics. Finally, we provide various explicit constructions of general Artinian Gorenstein rings corresponding to each stratum and discuss their lifting to higher dimension. These provide constructions of codimension four varieties, which include canonical surfaces, Calabi-Yau threefolds and Fano fourfolds. In the particular case of quaternary quartics, our results yield answers to questions posed by Geramita, Iarrobino-Kanev, and Reid.
Rational quartic spectrahedra in 3-space are semialgebraic convex subsets in $\mathbb{R}^3$ of semidefinite, real symmetric $(4 \times 4)$-matrices, whose boundary admits a rational parameterization. The Zariski closure in $\mathbb{C}\mathbb{P}^3$ of the boundary of a rational spectrahedron is a rational complex symmetroid. We give necessary conditions on the configurations of singularities of the corresponding real symmetroids in $\mathbb{R}\mathbb{P}^3$ of rational quartic spectrahedra. We provide an almost exhaustive list of examples realizing the configurations, and conjecture that the missing example does not occur.
We consider the problem of identifying a mixture of Gaussian distributions with same unknown covariance matrix by their sequence of moments up to certain order. Our approach rests on studying the moment varieties obtained by taking special secants to the Gaussian moment varieties, defined by their natural polynomial parametrization in terms of the model parameters. When the order of the moments is at most three, we prove an analogue of the Alexander-Hirschowitz theorem classifying all cases of homoscedastic Gaussian mixtures that produce defective moment varieties. As a consequence, identifiability is determined when the number of mixed distributions is smaller than the dimension of the space. In the two component setting we provide a closed form solution for parameter recovery based on moments up to order four, while in the one dimensional case we interpret the rank estimation problem in terms of secant varieties of rational normal curves.
We present a collection of research questions on cubic surfaces in 3-space. These questions inspired a collection of papers to be published in a special issue of the journal Le Matematiche. This article serves as the introduction to that 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.