Quartic Surface Research Articles

Algorithms to compute the topology of orientable real algebraic surfaces

We present constructive algorithms to determine the topological type of a non-singular orientable real algebraic projective surface S in the real projective space, starting from a polynomial equation with rational coefficients for S. We address this question when there exists a line in RP 3 not intersecting the surface, which is a decidable problem; in the case of quartic surfaces, when this condition is always fulfilled, we give a procedure to find a line disjoint from the surface. Our algorithm computes the homology of the various connected components of the surface in a finite number of steps, using as a basic tool Morse theory. The entire procedure has been implemented in Axiom.

Donaldson–Friedman construction and deformations of a triple of compact complex spaces, II

AbstractIn a previous paper of the same title the author gave a generalization of the constrution of Donaldson–Friedman, to prove the existence of twistor spaces of nCP2 with a special kind of divisors. In the present paper, we consider its equivariant version. When n = 3, this gives another proof of the existence of degenerate double solid with C*–action, and we show that the branch quartic surface is birational to an elliptic ruled surface. In case n ≥ 4, this yields new Moishezon twistor spaces with C*–action, which is shown to be the most degenerate ones among twistor spaces studied by Campana and Kreußler.

On quartics with three-divisible sets of cusps

We study the geometry and codes of quartic surfaces with many cusps. We apply Grobner bases to find examples of various configurations of cusps on quartics.

Counting rational points on cubic and quartic surfaces

Counting rational points on cubic and quartic surfaces

C2 quartic spline surface interpolation

This paper discusses the problem of constructing C2 quartic spline surface interpolation. Decreasing the continuity of the quartic spline to C2 offers additional freedom degrees that can be used to adjust the precision and the shape of the interpolation surface. An approach to determining the freedom degrees is given, the continuity equations for constructing C2 quartic spline curve are discussed, and a new method for constructing C2 quartic spline surface is presented. The advantages of the new method are that the equations that the surface has to satisfy are strictly row diagonally dominant, and the discontinuous points of the surface are at the given data points. The constructed surface has the precision of quartic polynomial. The comparison of the interpolation precision of the new method with cubic and quartic spline methods is included.

Corrigenda Arithmetic of Diagonal Quartic Surfaces, II

Corrigenda Arithmetic of Diagonal Quartic Surfaces, II

Galois points on normal quartic surfaces

Galois points on normal quartic surfaces

A C 1 -continuous formulation for 3D finite deformation frictional contact

A new 3D smooth triangular frictional node to surface contact element is developed using an abstract symbolical programming approach. The C1-continuous smooth contact surface description is based on the six quartic Bezier surfaces. The weak formulation and the penalty method are formulated for the description of large deformation frictional contact problems. The presented approach, based on a non-associated frictional law and elastic-plastic tangential slip decomposition, results into quadratic rate of convergence within the Newton-Raphson iteration loop. The frictional sliding path for the smooth, as well as the simple frictional node to surface contact element presented herein, is defined by the mapping of the current in the last converged configuration. Examples demonstrate the performance of symbolically developed contact elements, as well as the stability and more realistic contact description for the smooth elements in comparison with the simple ones.

Birational automorphisms of quartic Hessian surfaces

We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16 which embeds naturally in the even unimodular lattice II 1,25 of rank 26 and signature (1,25). The generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of Kondo. We shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface.

Galois points on quartic surfaces

Let S be a smooth hypersurface in the projective three space and consider a projection of S from P∈S to a plane H. This projection induces an extension of fields k(S)/k(H). The point P is called a Galois point if the extension is Galois. We study structures of quartic surfaces focusing on Galois points. We will show that the number of the Galois points is zero, one, two, four or eight and the existence of some rule of distribution of the Galois points.

Galois points on quartic surfaces

Let S be a smooth hypersurface in the projective three space and consider a projection of S from P∈S to a plane H. This projection induces an extension of fields k(S)/k(H). The point P is called a Galois point if the extension is Galois. We study structures of quartic surfaces focusing on Galois points. We will show that the number of the Galois points is zero, one, two, four or eight and the existence of some rule of distribution of the Galois points.

A Quartic Surface of Integer Hexahedra

We prove that there are infinitely many six sided polyhedra in R, each with four congruent trapezoidal faces and two congruent rectangular faces, so that the faces have integer sides and diagonals, and also the solid has integer length diagonals. The solutions are obtained from the integer points on a particular quartic surface. A long standing unsolved problem asks whether or not there can be a parallelipiped in R whose sides and diagonals have integer length. If one weakens the requirement and just asks for a six-sided polyhedron with quadrilateral faces, then one can find examples with integer length sides and diagonals. Peterson and Jordan [1], described a method for making these ‘perfect’ hexahedra. We review their method. One takes two congruent rectangles and places them in space so that they are parallel, with the bottom rectangle rotated ninety degrees from the position of the top rectangle. Now connect the sides of the two rectangles with four congruent trapezoids. The shape is a piecewise linear version of the placement of two cupped hands together, at ninety degrees in clapping position. The centers of the rectangles lie on a line perpendicular to the top and bottom faces. If the sides of the rectangle have lengths a, b then the diagonal has length c, where a + b = c. The parallel sides of the trapezoids are then also a, b. The slant side of the trapezoid is say e and its diagonal is d. It follows from Ptolemy’s Theorem that d = e + ab. Consider the trapezoid with base on the top rectangle of side a and other base on the bottom rectangle opposite that edge of side b, having the slant sides of length d. Its diagonal is of length f , and also it is the interior diagonal of the hexahedron; thus f 2 = d + ab. We shall refer to these as perfect hexahedra. Proposition 1. The simultaneous positive integer solutions to a + b = c, d = e + ab, f 2 = d + ab give the edge and diagonal lengths of a perfect hexahedron with two rectangular congruent opposite parallel faces, and four congruent trapezoidal faces. Peterson and Jordan asked if there are infinitely many such perfect hexahedra. They also gave several examples, including the small example a = 8, b = 15, c = 17, e = 7, d = 13, f = 17 and asked if it is the

Compactification of a family of quartic del Pezzo surfaces

Compactification of a family of quartic del Pezzo surfaces

Compactification of a family of quartic del Pezzo surfaces

Compactification of a family of quartic del Pezzo surfaces

Selmer groups and zero-cycles on the Fermat quartic surface

Selmer groups and zero-cycles on the Fermat quartic surface

Evidence of Fermi resonance in core-ionized methane

A full quartic potential energy surface is determined for core-ionized methane and used to investigate coupling between vibrational modes. A strong Fermi resonance is found between the first excited state of the symmetric stretching mode ν1′ and a doubly excited bending mode, whereas the corresponding interaction is less pronounced for v1′=2. In terms of the carbon 1s photoelectron spectrum of methane, the net effect of the mode coupling is to reduce the apparent contribution from anharmonicity to peak positions. The contribution from anharmonicity to the intensity of each peak is dominated by cubic and quartic terms in the symmetric stretching coordinate, and remains significant. This resolves a paradox pointed out in a recent experimental work [Carroll et al., Phys. Rev. A 59, 3386 (1999)].

Arithmetic of diagonal quartic surfaces, II

Arithmetic of diagonal quartic surfaces, II

Another Description of Certain Quartic Double Solids

The aim of this short note is to give an explicit description of birationally equivalent models of quartic double solids with at least one node. A quartic double solid is a double covering of ℙ3 branched along a quartic surface. The nodes of the double solid are in bijection with the nodes of the quartic surface B ⊂ ℙ3. Using the projection ℙ 3 → ℙ2, whose center is a node of B, we obtain a birational map to a conic bundle over ℙ2 . We shall explicitly describe a birational equivalence between such a conic bundle and a cubic hypersurface in ℙ4. We obtain this by projecting the cubic from a smooth point.

Tangent Conics at Quartic Surfaces and Conics in Quartic Double Solids

For a quartic double solid we study the parameter space of conics (i.e. of smooth rational curves C ⊂ Z such that C · φ*O(1) = 2). This parameter space is naturally fibred (with disconnected fibres) over Pˇ3. We study the monodromy of the fibres and determine this way the irreducible components of the parameter space.

On the normal bundle of curves on nodal quartic surfaces

Let k be an algebraically closed field, char k = 0. Let C be an irreducible nonsingular curve such that 2C = S ⋂ F, where S and F are two surfaces and all the singularities of F are rational double points (if any). We prove that C can never pass through rational singularities of types A 2n n∈N, E6 and E8. We give conditions for C to pass through rational singularities of types. A 2k+1 k∈Z+ Dn n≥4 and E7, (0.8).