A computational approach to Lüroth quartics

A plane quartic curve is called Lüroth if it contains the ten vertices of a complete pentalateral. White and Miller constructed in 1909 a covariant quartic fourfold, associated to any plane quartic. We review their construction and we show how it gives a computational tool to detect if a plane quartic is Lüroth. As a byproduct, we show that the 28 bitangents of a general plane quartic correspond to 28 singular points of the associated White–Miller quartic fourfold.     

The Plane Sextic Curve Fixed by A6

Plane curves with non-trivial collineation groups are rare: those of low order are thus interesting, and often exhibit special geometric features. The largest primitive plane group is A₆. It is known by standard algebraic means that this fixes a sextic curve Ω. The present paper constructs Ω geometrically, and speedily obtains its most significant geometric property: its 72 inflexions lie by pairs on 36 biflexional tangents. There is no standard technique for determining the multiplicities of bitangents of plane curves. For Ω we show that each biflexional tangent counts 4-fold as a bitangent, and identify the other 180 ordinary bitangents. A brief comparison of the geometric properties of Ω with those of Klein’s quartic curve is given.     

Generic algebras

The paper studies generic commutative and anticommutative algebras of a fixed dimension, their invariants, covariants and algebraic properties (e.g., the structure of subalgebras). In the case of 4-dimensional anticommutative algebras a construction is given that links the associated cubic surface and the 27 lines on it with the structure of subalgebras of the algebra. The rationality of the corresponding moduli variety is proved. In the case of 3-dimensional commutative algebras a new proof of a recent theorem of Katsylo and Mikhailov about the 28 bitangents to the associated plane quartic is given. The research was supported by Grant # MQZ300 from the ISF and Russian Government.     

Special subvarieties of EPW sextics

We study the geometry of EPW sextics in order to produce special subvarieties. In particular we exhibit a (singular) Enriques surface and we compute its class in the Chow ring of the sextic. In order to do this, we produce an explicit degeneration of double EPW sextics to a Hilbert scheme of two points on a quartic surface, both in the smooth and in the singular case (keeping the singularities). The fixed locus of the covering involution on the double EPW sextic degenerates to the surface of bitangents to the quartic, which can be shown to be birational to an Enriques surface, provided the quartic acquires enough nodes. This construction is used in Ferretti (Algebra Number Theory, 2009a) as a starting point to prove a conjecture of Beauville and Voisin on the Chow ring of irreducible symplectic varieties, in the particular case of a very general double EPW sextic.     

Recovering plane curves of low degree from their inflection lines and inflection points

In this paper we consider the following problem: is it possible to recover a smooth plane curve of degree d ≥ 3 from its inflection lines? We answer the posed question positively for a general smooth plane quartic curve, making the additional assumption that also one inflection point is given, and for any smooth plane cubic curve.     

Elementary transformations of pfaffian representations of plane curves

Let C be a smooth curve in P² given by an equation F=0 of degree d. In this paper we consider elementary transformations of linear pfaffian representations of C. Elementary transformations can be interpreted as actions on a rank 2 vector bundle on C with canonical determinant and no sections, which corresponds to the cokernel of a pfaffian representation. Every two pfaffian representations of C can be bridged by a finite sequence of elementary transformations. Pfaffian representations and elementary transformations are constructed explicitly. For a smooth quartic, applications to Aronhold bundles and theta characteristics are given.     

The local–global principle for symmetric determinantal representations of smooth plane curves

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local–global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi–Brauer varieties and mod 2 Galois representations, and prove that the local–global principle holds for conics and cubics. We also construct counterexamples to the local–global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics.     

Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones

We prove that if there exists a bi-Lipschitz homeomorphism (not necessarily subanalytic) between two subanalytic sets, then their tangent cones are bi-Lipschitz homeomorphic. As a consequence of this result, we show that any Lipschitz regular complex analytic set, i.e., any complex analytic set which is locally bi-Lipschitz homeomorphic to an Euclidean ball must be smooth. Finally, we give an alternative proof of S. Koike and L. Paunescu’s result about the bi-Lipschitz invariance of directional dimensions of subanalytic sets.     

Twelve points on the projective line, branched covers, and rational elliptic fibrations

The following divisors in the space of twelve points on are actually the same: the possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves); degree 12 binary forms that can be expressed as a cube plus a square; the locus of the twelve tangents to a smooth plane quartic from a general point of the plane; the branch locus of a degree 4 map from a hyperelliptic genus 3 curve to ; the branch locus of a degree 3 map from a genus 4 curve to induced by a theta-characteristic; and several more. The corresponding moduli spaces are smooth, but they are not all isomorphic; some are finite étale covers of others. We describe the web of interconnections among these spaces, and give monodromy, rationality, and Prym-related consequences. Enumerative consequences include: (i) the degree of this locus is 3762 (e.g. there are 3762 rational elliptic fibrations with nodes above 11 given general points of the base); (ii) if is a cover as in , then there are 135 different such covers branched at the same points; (iii) the general set of 12 tangent lines that arise in turn up in 120 essentially different ways. Some parts of this story are well known, and some other parts were known classically (to Zeuthen, Zariski, Coble, Mumford, and others). The unified picture is surprisingly intricate and connects many beautiful constructions, including Recillas' trigonal construction and Shioda's -Mordell-Weil lattice. Received November 3, 1999 / Published online February 5, 2001     

Counting bitangents with stable maps

This paper is an elementary introduction to the theory of moduli spaces of curves and maps. As an application to enumerative geometry, we show how to count the number of bitangent lines to a projective plane curve of degree d by doing intersection theory on moduli spaces.     

The Hilbert function of curves on certain smooth quartic surfaces

The Riemann-Roch problem for divisors on a smooth surface in ³ is studied. This problem is solved for some smooth quartic surfaces which are called Mori quartics; as a consequence the Hilbert function of any integral curve on a Mori quartic is determined.     

An unexpected appearance of Steiner's hypocycloid

J. de Cicco [1939] observed that two parabolas must touch each other if they have parallel axes, while one parabola touches the three sides of a given triangle and the other passes through the midpoints of those sides. Coxeter [1983] showed that the locus of the point of contact of the two parabolas, if the triangle is kept fixed while the common axial direction varies, is a rational cubic curve. In a subsequent paper, Coxeter investigated other aspects of this cubic (Coxeter [1985]).De Cicco's theorem, viewed as a result in the projective plane, can be dualized in a natural way. The cubic then becomes a set of lines, enveloping a curve of class three. We shall show that this curve is a quartic curve with three cusps, which is projectively equivalent to Steiner's well-known hypocycloid.     

Distance sets that are a shift of the integers and Fourier basis for planar convex sets

The aim of this paper is to prove that if a planar set A has a difference set Δ(A) satisfying Δ(A) ? ?⁺ + s for suitable s then A has at most 3 elements. This result is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. Further, we prove that if A is a set of exponentials mutually orthogonal with respect to any symmetric convex set K in the plane with a smooth boundary and everywhere non-vanishing curvature, then #(A ∩ [?q, q]²) ≦ C(K) _q where C(K) is a constant depending only on K. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from [8] and [9] that if K is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then L ²(K) does not possess an orthogonal basis of exponentials.     

关于平面四次Bézier曲线的拐点与奇点

在计算几何中,已给出了三次Bezier曲线的保凸性的充要条件,并进行了几何解释。本文则是导出形式简洁的拐点和奇点方程并对四次Bezier曲线的拐点和奇点的分布进行讨论。按Bezier曲线的拐点个数进行分类,还得到了四次Bezier曲线有奇点的充分必要条件,并给出几个数值实例,实例说明,不但非凸的单纯特征多角形可以有凸的Bezier曲线段,而且非单纯特征多角形也可以有凸的Bezier曲线段。四次Bezier曲线的奇点和拐点是可以共存的。     

An anisotropic area-preserving flow for convex plane curves

This paper will deal with an anisotropic area-preserving flow which keeps the convexity of the evolving curve and the limiting curve converges to a homothety of a symmetric smooth strictly convex plane curve.     

A quadratically convergent local algorithm on minimizing sums of the largest eigenvalues of a symmetric matrix

In this paper, we consider the problem on minimizing sums of the largest eigenvalues of a symmetric matrix which depends on the decision variable affinely. An important application of this problem is the graph partitioning problem, which arises in layout of circuit boards, computer logic partitioning, and paging of computer programs. Given 0, we first derive an optimality condition which ensures that the objective function is within error bound of the solution. This condition may be used as a practical stopping criterion for any algorithm solving the underlying problem. We also show that, in a neighborhood of the minimizer, the optimization problem can be equivalently formulated as a smooth constrained problem. An existing algorithm on minimizing the largest eigenvalue of a symmetric matrix is shown to be applicable here. This algoritm enjoys the property that if started close enough to the minimizer, then it will converge quadratically. To implement a practical algorithm, one needs to incorporate some technique to generate a good starting point. Since the problem is convex, this can be done by using an algorithm for general convex optimization problems (e.g., Kelley's cutting plane method or ellipsoid methods), or an algorithm specific for the optimization problem under consideration (e.g., the algorithm developed by Cullum, Donath, and Wolfe). Such union ensures that the overall algorithm has global convergence with quadratic rate. Finally, the results presented in this paper are readily extended on minimizing sums of the largest eigenvalues of a Hermitian matrix.Some of results in this paper were given in [19] without proofs.     

Combinatorial analysis of proofs in projective and affine geometry

The axioms of projective and affine plane geometry are turned into rules of proof by which formal derivations are constructed. The rules act only on atomic formulas. It is shown that proof search for the derivability of atomic cases from atomic assumptions by these rules terminates (i.e., solves the word problem). This decision method is based on the central result of the combinatorial analysis of derivations by the geometric rules: The geometric objects that occur in derivations by the rules can be restricted to those known from the assumptions and cases. This “subterm property” is proved by permuting suitably the order of application of the geometric rules. As an example of the decision method, it is shown that there cannot exist a derivation of Euclid’s fifth postulate if the rule that corresponds to the uniqueness of the parallel line construction is taken away from the system of plane affine geometry.     

Symbolic Computation for Rankin-Cohen Differential Algebras: A Case Study

Don Zagier defined a “Rankin-Cohen algebra”, motivated by the study of differential operators that send modular forms to modular forms. We devised an algorithm that computes the result of the differentiation given by the modular forms that correspond to higher-order Wronskians over Klein’s quartic curve, which are modular forms of arbitrarily high degree canonically attached to the curve; this tool is potentially useful for finding commutative rings of differential operators.     

THE SINE TRANSFORM OPERATOR IN THE BANACH SPACE OF SYMMETRIC MATRICES AND ITS APPLICATION IN IMAGE RESTORATION

In this paper, we study an operator s which maps every n-by-n symmetric matrix A, to a matrix s(A_n) that minimizes || B_n-A_n || F over the set of all matrices B_n, that can be diagonalized by the sine transform. The matrix s(A_n), called the optimal sine transform preconditioner, is defined for any n-by-n symmetric matrices A_n. The cost of constructing s(A_n) is the same as that of optimal circulant preconditioner c(A_n) which is defined in [8], The s(A_n) has been proved in [6] to be a good preconditioner in solving symmetric Toeplitz systems with the preconditioned conjugate gradient (PCG) method. In this paper, we discuss the algebraic and geometric properties of the operator s, and compute its operator norms in Banach spaces of symmetric matrices. Some numerical tests and an application in image restoration are also given.     

Symmetric Orientations of Dividing T-curves

In 1974, Rokhlim introduced complex orientations for nonsingular real algebraic plane projective curves of type I. Here we give a definition of symmetric orientations and of "type" for T-curves which are PL-curves constructed using a combinatorial method called T-construction. An important aspect of T-construction is that, under particular conditions, the constructed T-curve has the isotopy type of a nonsingular real algebraic plane projective curve. T-construction is in fact a particular case of the method of construction of real algebraic projective varieties due to O. Ya. Viro. We prove that if an algebraic curve is associated to a T-curve by the Viro process, then the type of the T-curve coincides with the type of the algebraic curve and its symmetric orientations are complex orientations as defined by Rokhlin. The main result of this paper is the classification theorem for T-curves of type I.