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Nodal Surface
In algebraic geometry, a nodal surface is a surface in (usually complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...) projective space whose only singularities are nodes. A major problem about them is to find the maximum number of nodes of a nodal surface of given degree. The following table gives some known upper and lower bounds for the maximal number of nodes on a complex surface of given degree. In degree 7, 9, 11, and 13, the upper bound is given by , which is better than the one by . See also * Algebraic surface References * * * *{{citation , mr=3124329 , doi=10.1016/j.crma.2013.09.009 , last=Escudero , first=Juan García , title=On a family of complex algebraic surfaces of degree 3''n'' , journal=C. R. Math. Acad. Sci. Paris , volume=351 , year=2013 , i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Projective Space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the ''complex'' lines through the origin of a complex Euclidean space (see below for an intuitive account). Formally, a complex projective space is the space of complex lines through the origin of an (''n''+1)-dimensional complex vector space. The space is denoted variously as P(C''n''+1), P''n''(C) or CP''n''. When , the complex projective space CP1 is the Riemann sphere, and when , CP2 is the complex projective plane (see there for a more elementary discussion). Complex projective space was first introduced by as an instance of what was then known as the "geometry of position", a notion originally due to Lazare Carnot, a kind of synthetic geometry that included other projective geometries as well. Sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Point Of An Algebraic Variety
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth. Definition A plane curve defined by an implicit equation :F(x,y)=0, where is a smooth function is said to be ''singular'' at a point if the Taylor series of has order at least at this point. The reason for this is that, in differential calculus, the tangent at the point of such a curve is defined by the equation :(x-x_0)F'_x(x_0,y_0) + (y-y_0)F'_y(x_0,y_0)=0, whose left-hand side is the term of degree one of the Taylor expansion. Thus, if this term is zero, the tangent may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conical Surface
In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the ''apex'' or ''vertex'' — and any point of some fixed space curve — the ''directrix'' — that does not contain the apex. Each of those lines is called a ''generatrix'' of the surface. Every conic surface is ruled and developable. In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and pass through a point of some fixed space curve. (In some cases, however, the two nappes may intersect, or even coincide with the full surface.) Sometimes the term "conical surface" is used to mean just one nappe. If the directrix is a circle C, and the apex is located on the circle's ''axis'' (the line that contains the center of C and is perpendicular to its plane), one obtains the ''right circula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley's Nodal Cubic Surface
In algebraic geometry, the Cayley surface, named after Arthur Cayley, is a cubic nodal surface in 3-dimensional projective space with four conical points. It can be given by the equation : wxy+ xyz+ yzw+zwx =0\ when the four singular points are those with three vanishing coordinates. Changing variables gives several other simple equations defining the Cayley surface. As a del Pezzo surface of degree 3, the Cayley surface is given by the linear system of cubics in the projective plane passing through the 6 vertices of the complete quadrilateral. This contracts the 4 sides of the complete quadrilateral to the 4 nodes of the Cayley surface, while blowing up its 6 vertices to the lines through two of them. The surface is a section through the Segre cubic. The surface contains nine lines, 11 tritangents and no double-sixes. A number of affine forms of the surface have been presented. Hunt uses (1-3 x-3y-3z)(xy+xz+yz)+6xyz = 0 by transforming coordinates (u_0, u_1, u_2, u_3) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kummer Surface
In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb^3 with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variety of a smooth hyperelliptic curve of genus 2; i.e. a quotient of the Jacobian by the Kummer involution ''x'' ↦ −''x''. The Kummer involution has 16 fixed points: the 16 2-torsion point of the Jacobian, and they are the 16 singular points of the quartic surface. Resolving the 16 double points of the quotient of a (possibly nonalgebraic) torus by the Kummer involution gives a K3 surface with 16 disjoint rational curves; these K3 surfaces are also sometimes called Kummer surfaces. Other surfaces closely related to Kummer surfaces include Weddle surfaces, wave surfaces, and tetrahedroids. Geometry of the Kummer surface Singular quartic surfaces and the double plane model Let K\subset\mathbb^3 be a quartic surface with an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Togliatti Surface
In algebraic geometry, a Togliatti surface is a nodal surface of degree five with 31 Node (algebraic geometry), nodes. The first examples were constructed by . proved that 31 is the maximum possible number of nodes for a surface of this degree, showing this example to be optimal. See also *Barth surface *Endrass surface *Sarti surface *List of algebraic surfaces References *. *. External links * * Algebraic surfaces Complex surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barth Sextic
__NOTOC__ In algebraic geometry, a Barth surface is one of the complex nodal surfaces in 3 dimensions with large numbers of double points found by . Two examples are the Barth sextic of degree 6 with 65 double points, and the Barth decic of degree 10 with 345 double points. For degree 6 surfaces in P3, showed that 65 is the maximum number of double points possible. The Barth sextic is a counterexample to an incorrect claim by Francesco Severi in 1946 that 52 is the maximum number of double points possible. Informal accounting of the 65 ordinary double points of the Barth Sextic The Barth Sextic may be visualized in three dimensions as featuring 50 finite and 15 infinite ordinary double points (nodes). Referring to the figure, the 50 finite ordinary double points are arrayed as the vertices of 20 roughly tetrahedron, tetrahedral shapes oriented such that the bases of these four-sided "outward pointing" shapes form the triangular faces of a regular icosidodecahedron. To these 3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Labs Septic
In mathematics, the Labs septic surface is a degree-7 ( septic) nodal surface with 99 nodes found by . As of 2015, it has the largest known number of nodes of a degree-7 surface, though this number is still less than the best known upper bound of 104 nodes given by .The upper bound in degree 7 given by is 106. See also *Barth surface *Endrass surface *Sarti surface *Togliatti surface In algebraic geometry, a Togliatti surface is a nodal surface of degree five with 31 Node (algebraic geometry), nodes. The first examples were constructed by . proved that 31 is the maximum possible number of nodes for a surface of this degree, ... References * * * External links * Algebraic surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endraß Surface
In algebraic geometry, an Endrass surface is a nodal surface of degree 8 with 168 real nodes, found by . , it remained the record-holder for the most number of real nodes for its degree; however, the best proven upper bound, 174, does not match the lower bound given by this surface. See also *Barth surface *Sarti surface In algebraic geometry, a Sarti surface is a degree-12 nodal surface with 600 nodes, found by . The maximal possible number of nodes of a degree-12 surface is not known (as of 2015), though Yoichi Miyaoka showed that it is at most 645. Sarti has ... * Togliatti surface References {{reflist Algebraic surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barth Decic
__NOTOC__ In algebraic geometry, a Barth surface is one of the complex nodal surfaces in 3 dimensions with large numbers of double points found by . Two examples are the Barth sextic of degree 6 with 65 double points, and the Barth decic of degree 10 with 345 double points. For degree 6 surfaces in P3, showed that 65 is the maximum number of double points possible. The Barth sextic is a counterexample to an incorrect claim by Francesco Severi in 1946 that 52 is the maximum number of double points possible. Informal accounting of the 65 ordinary double points of the Barth Sextic The Barth Sextic may be visualized in three dimensions as featuring 50 finite and 15 infinite ordinary double points (nodes). Referring to the figure, the 50 finite ordinary double points are arrayed as the vertices of 20 roughly tetrahedral shapes oriented such that the bases of these four-sided "outward pointing" shapes form the triangular faces of a regular icosidodecahedron. To these 30 icosidode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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