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Kristian Ranestad

Publications and source records attributed to Kristian Ranestad.

At least 55 records · Page 3Linked to original sources
arXiv2009

Linear precision for toric surface patches

We classify the homogeneous polynomials in three variables whose toric polar linear system defines a Cremona transformation. This classification also includes, as a proper subset, the classification of toric surface patches from geometric modeling which have linear precision. Besides the well-known tensor product patches and Bézier triangles, we identify a family of toric patches with trapezoidal shape, each of which has linear precision. Bézier triangles and tensor product patches are special cases of trapezoidal patches.

math.AG↗
arXiv2008

The Algebraic Degree of Semidefinite Programming

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.

math.OC↗
arXiv2008

Algebraic Degree of Polynomial Optimization

Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on optimizers, and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials. We also give a general formula for the algebraic degree of the optimal coordinates. The derivation of the algebraic degree is equivalent to counting the number of all complex critical points. As special cases, we obtain the algebraic degrees of quadratically constrained quadratic programming (QCQP), second order cone programming (SOCP) and $p$-th order cone programming (pOCP), in analogy to the algebraic degree of semidefinite programming.

math.OC↗
arXiv2007

Classification of rational surfaces of degree 11 and sectional genus 11 in P^4

We use the BGG-correspondence to show that there are at most three possible Hilbert functions for smooth rational surfaces of degree 11 and sectional genus 11. Surfaces with one of these Hilbert functions have been classified by Popescu. The classification for a second one is done in this paper. For the third Hilbert function the classification is still open.

math.AG↗
arXiv2006

Addendum to "K3-surfaces of genus 8 and varieties of sums of powers of cubic fourfolds"

In this note, which is an addendum to the e-print math.AG/9810121, we prove that the variety VSP(F,10) of presentations of a general cubic form F in 6 variables as a sum of 10 cubes is a smooth symplectic 4-fold, which is deformation equivalent to the Hilbert square of a K3 surface of genus 8 but different from the family of lines on a cubic 4-fold. This provides a new geometric construction of a compact complex symplectic fourfold, different from a Hilbert square of a K3 surface, a generalized Kummer 4-fold, the variety of lines on a cubic 4-fold and the recent examples of O'Grady (see Duke Math. J. 134, no. 1 (2006), 99-137).

math.AG↗
arXiv2006

K3-surfaces of genus 8 and varieties of sums of powers of cubic fourfolds

The main outcome of this paper is that the variety VSP(F,10) of presentations of a general cubic form F in 6 variables as a sum of 10 cubes is a smooth symplectic 4-fold obtained a deformation of the Hilbert square of a K3 surface of genus 8. After publishing it in Trans. Am. Math. Soc. 353, No.4, 1455-1468 (2001), it was noted to us by Eyal Markman that in Theorem 3.17 we conclude without proof that VSP(F,10) should be the 4-fold of lines on another cubic 4-fold. We correct this in the e-print "Addendum to K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds" (math.AG/0611533), where we establish that the general VSP(F,10) is in fact a new symplectic 4-fold different from the family of lines on a cubic 4-fold.

math.AG↗
arXiv2005

The abelian fibration on the Hilbert cube of a K3 surface of genus 9

In this paper we construct an abelian fibration over ${\bf P}^3$ on the Hilbert cube of the primitive K3 surface of genus 9. After the abelian fibration constructed by Mukai on the Hilbert square on the primitive K3 surface S of genus 5, this is the second example where the abelian fibration on such Hilb_n(S) is directly constructed. Our example is also the first known abelian fibration on a Hilbert scheme Hilb_n(S) of a primitive K3 surface S which is not the Hilbert square of S; the primitive K3 surfaces on the Hilbert square of which such a fibration exists are known by a recent result of Hassett and Tschinkel.

math.AG↗
arXiv2005

Schubert Unions in Grassmann Varieties

We study subsets of Grassmann varieties G(l,m) over a field F, such that these subsets are unions of Schubert cycles, with respect to a fixed flag. We study the linear spans of, and in case of positive characteristic, the number of points on such unions over finite fields. Moreover we study a geometric duality of such unions, and give a combinatorial interpretation of this duality. We discuss the maximum number of points over a finite field for the Schubert unions of a given spanning dimension, and we give some applications to coding theory. We define Schubert union codes, and study the parameters and support weights of these codes and of the well-known Grassmann codes.

math.AG↗
arXiv2004

Cremona Convexity, Frame Convexity, and a Theorem of Santaló

In 1940, Luis Santaló proved a Helly-type theorem for line transversals to boxes in R^d. An analysis of his proof reveals a convexity structure for ascending lines in R^d that is isomorphic to the ordinary notion of convexity in a convex subset of R^{2d-2}. This isomorphism is through a Cremona transformation on the Grassmannian of lines in P^d, which enables a precise description of the convex hull and affine span of up to d ascending lines: the lines in such an affine span turn out to be the rulings of certain classical determinantal varieties. Finally, we relate Cremona convexity to a new convexity structure that we call frame convexity, which extends to arbitrary-dimensional flats.

math.MG↗
arXiv2004

Phylogenetic Algebraic Geometry

Phylogenetic algebraic geometry is concerned with certain complex projective algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover classical geometric objects, such as toric and determinantal varieties and their secant varieties, but larger trees lead to new and largely unexplored territory. This paper gives a self-contained introduction to this subject and offers numerous open problems for algebraic geometers.

math.AG↗
arXiv2003

Degenerations of (1,7)-polarized abelian surfaces

The moduli space of (1,7)-polarized abelian surfaces with a level structure was shown by Manolache and Schreyer to be rational with compactification a Fano 3-fold of genus 12 which is also a compactification of the variety of powersum presentations of the Klein quartic curve. In this paper the possible degenerations of the abelian surfaces are classified in terms of degenerations of powersum presentations. In particular, the surface degenerations into fourteen planes are recovered. These were independently discovered by Alfio Marini using degenerations of twisted cubic curves.

math.AG↗
arXiv2002

Geometry of the Lagrangian Grassmannian Sp(3)/U(3) with applications to Brill-Noether loci

The geometry of Sp(3)/U(3) as a subvariety of Gr(3,6) is explored to explain several examples given by Mukai of non-abelian Brill-Noether loci, and to give some new examples. These examples identify Brill-Noether loci of vector bundles on linear sections of the Lagrangian Grassmannian Sp(3)/U(3) with orthogonal linear sections of the dual variety and vice versa. A main technical result of independent interest is the fact that any nodal hyperplane section of the Lagrangian Grassmannian projected from the node is a linear section of the Grassmannian Gr(2,6).

math.AG↗
arXiv2002

Irregular elliptic surfaces of degree 12 in the projective fourspace

So far only a few families of smooth irregular surfaces are known to exist in P^4 up to pullbacks by suitable finite morphisms from P^4 onto P^4 itself. In this paper we present two different constructions of irregular smooth minimal elliptic surfaces of degree 12 in P^4. The first is a monad construction while the other uses liaison. The family constructed via liaison includes the surfaces of the first family as a special case.

math.AG↗
arXiv1998

Abelian surfaces with two plane cubic curve fibrations and Calabi-Yau threefolds

(1,d)-polarized abelian surfaces in P^(d-1) with two plane cubic curve fibrations lie in two elliptic P^2-scrolls. The union of these scrolls form a reducible Calabi-Yau 3-fold. In this paper we show that this occurs when d<10 and analyse the family of such surfaces and 3-folds in detail when d=6. In particular, the reducible Calabi-Yau 3-folds deform in that case to irreducible ones with non-normal singularities.

math.AG↗
arXiv1997

On surfaces in P^6 with no trisecant lines

We construct examples of smooth surfaces S in P^6 with no trisecant lines. This list includes examples of surfaces not cut out by quadrics. We prove that unless S has a finite number of disjoint $(-1)$-lines, and each one meets some other line L on the surface with L^2\leq -2, and S is not an inner projection from P^7 the surface belongs to the list of examples.

alg-geom↗
arXiv1997

Syzygies of Abelian and Bielliptic Surfaces in P^4

So far only six families of smooth irregular surfaces are known to exist in P^4 (up to pullbacks by suitable finite covers of P^4). These are the elliptic quintic scrolls, the minimal abelian and bielliptic surfaces (of degree 10), two different families of non-minimal abelian surfaces of degree 15, and one family of non-minimal bielliptic surfaces of degree 15. The main purpose of the paper is to describe the structure of the Hartshorne-Rao modules and the syzygies for each of these smooth irregular surfaces in P^4, providing at the same time a unified construction method (via syzygies) for these families of surfaces.

alg-geom↗