R sup 2 sup are 2 variables that parametrize the image. Be careful that a parametric surface need .... The number g of tori involved is called the genus of the surface. The sphere and the torus have Euler ... sum of k of them is nowrap 2 &minus k . It follows that a closed surface is determined, up ... compact surface is classified by the number of boundary components and the genus of the corresponding .... The genus of a compact surface is defined as the genus of the corresponding closed surface ... surface. The unique compact orientable surface of genus g and with k boundary components is often denoted ... Riemann surface s, i.e., compact complex 1 manifolds. Note that the 2 sphere and the torus are both ... curvature K over the entire surface S is determined by the Euler characteristic math int S K dA 2 ... geometry of surfaces , algebraic surface , and Surface disambiguation . Image Saddle pt.jpg thumb 225px right An open surface with X , Y , and Z contours shown. In mathematics , specifically in topology , a surface is a two dimensional topological manifold . The most familiar examples are those ... 3 sup &mdash for example, the surface of a ball . On the other hand, there are surfaces, such as the Klein ... singularity theory singularities or self intersections. To say that a surface is two dimensional ... system is defined. For example, the surface of the Earth is ideally a two dimensional sphere , and latitude ... meridian . The concept of surface finds application in physics , engineering , computer graphics ... is the flow of air along its surface. Definitions and first examples A topological surface is a nonempty ... open set open subset of the Euclidean plane E sup 2 sup . Such a neighborhood, together with the corresponding ... generally, a topological surface with boundary is a Hausdorff space Hausdorff topological space in which ... open set open subset of the upper half plane H sup 2 sup . These homeomorphisms are also known as coordinate charts . The boundary of the upper half plane is the x axis. A point on the surface mapped ... more details
Image Bolza surface projection.png 300px right thumb Perspective projection of math y 2 x 5 x math in math mathbb C 2 math . In mathematics , the Bolza surface , alternatively, complex algebraic Bolza curve named after Oskar Bolza , is a compact Riemann surface of genus mathematics genus2 with the highest possible order of the conformal map conformal automorphism group in this genus, namely 48. An affine model for the Bolza surface can be obtained as the locus of the equation math y 2 x 5 x math in math mathbb C 2 math . The Bolza surface is the smooth completion of the affine curve. Of all genus2 hyperbolic surfaces, the Bolza surface has the highest systolic geometry systole . As a hyperelliptic surface, it arises as the ramified double cover of the Riemann sphere, with ramification locus at the six vertices of a regular octahedral symmetry octahedron inscribed in the sphere, as can be readily seen from the equation above. Triangle surface The Bolza surface is a 2,3,8 triangle surface see Schwarz triangle . More specifically, the Fuchsian group defining the Bolza surface is a subgroup ..., which has an abstract presentation in terms of generators math s 2, s 3, s 8 math and relations math s 22 s 3 3 s 8 8 1 math as well as math s 2 s 3 s 8 math . The Fuchsian group defining the Bolza surface is also a subgroup of the 3,3,4 triangle group , which is a subgroup of index 2 ... pi 2 , tfrac pi 3 , tfrac pi 8 math . More specifically, it is a subgroup of the Index of a subgroup ... and Reid, the quaternion algebra can be taken to be the algebra over math mathbb Q sqrt 2 math generated as an associative algebra by generators i,j and relations math i 2 3, j 2 sqrt 2 , ij ji, math with an appropriate choice of an order ring theory order . See also Klein quartic Macbeath surface ... for CAT 0 metrics in genus two journal Pacific Journal of Mathematics Pacific J. Math. volume ... geometry navbox Category Riemann surfaces Category Systolic geometry fr Surface de Bolza sl Bolzova ... more details
for spaces of valuations Zariski Riemann surface In algebraic geometry , a branch of mathematics , a Zariski surface is a surface over a field mathematics field of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane to the surface. In particular, all Zariski surfaces are unirational . They were named by Piotr Blass after Oscar Zariski who used them in 1958 to give examples of unirational surfaces in characteristic p 0 that are not rational. In characteristic 0 by contrast, Castelnuovo s theorem implies that all unirational surfaces are rational. Zariski surfaces are birational to surfaces in affine space affine 3 space A sup 3 sup defined by irreducible polynomial s of the form math z p f x, y . math The following problem posed by Oscar Zariski in 1971 is still open let p 5, let S be a Zariski surface with vanishing geometric genus. Is S necessarily a rational surface? See also List of algebraic surfaces References Citation last1 Zariski first1 Oscar author1 link Oscar Zariski title On Castelnuovo s criterion of rationality p sub a sub P sub 2 sub 0 of an algebraic surface url http projecteuclid.org euclid.ijm 1255454536 id MathSciNet id 0099990 year 1958 journal Illinois Journal of Mathematics issn 0019 2082 volume 2 pages 303 315 Category algebraic varieties Category algebraic surfaces ... more details
Image Double torus illustration.png right thumb A genus2surface. In mathematics, a genus2surface also known as a double torus or two holed torus is a surface formed by the connected sum of two torus tori . That is to say, from each of two tori the interior of a disk is removed, and the boundaries of the two disks are identified glued together , forming a double torus. This is the simplest case of the connected sum of n tori. A connected sum of tori is an example of a two dimensional manifold mathematics manifold . According to the surface classification theorem for 2 manifolds, every compact space compact connected space connected 2 manifold is either a sphere, a connected sum of tori, or a connected sum of real projective plane s. Double torus knot s are studied in knot theory . Example The Bolza surface is the most symmetric hyperbolic surface of Genus mathematics genus2. See also Triple torus References James R. Munkres, Topology, Second Edition , Prentice Hall, 2000, ISBN 0 13 181629 2. William S. Massey, Algebraic Topology An Introduction , Harbrace, 1967. External links MathWorld title Double Torus urlname DoubleTorus Category Topology eo Duopa toro ... more details
of the full triangle group. Examples The Hurwitz surface of least genus is the Klein quartic of genus ... 168 2 sup 2 sup 3 7, which is a simple group or order 336 if one allows orientation reversing isometries . The next possible genus is 7, possessed by the Macbeath surface , with automorphism ...File Uniform tiling 73 t2.png thumb Every Hurwitz surface has a triangulation as a quotient of the order 7 triangular tiling , with the automorphisms of the triangulation equaling the Riemannian and algebraic automorphisms of the surface. In Riemann surface theory and hyperbolic geometry , a Hurwitz surface , named after Adolf Hurwitz , is a compact Riemann surface with precisely 84 g &minus 1 automorphisms, where g is the Genus mathematics genus of the surface. This number is maximal by virtue of Hurwitz s theorem on automorphisms Harv Hurwitz 1893 . They are also referred to as Hurwitz curves , interpreting them as complex algebraic curves complex dimension 1 real dimension 2 . The Fuchsian group of a Hurwitz surface is a finite Index of a subgroup index torsionfree normal subgroup of the ordinary 2,3,7 triangle group . The finite quotient group is precisely the automorphism group. Automorphisms of complex algebraic curves are orientation preserving automorphisms of the underlying real surface if one allows orientation reversing isometries, this yields a group twice as large, of order ... , which is index 2. The group of complex automorphisms is a quotient of the ordinary orientation preserving ... 2 sup 2 sup 3 sup 2 sup 7 if one includes orientation reversing isometries, the group is of order 1,008. An interesting phenomenon occurs in the next possible genus, namely 14. Here there is a triple ... 1092 2 3 7 13 . The explanation for this phenomenon is arithmetic. Namely, in the ring ... www.heldermann.de BAG bag44.htm 44 year 2003 pages 413 430 title The Riemann Surface of a Uniform ... http www.heldermann.de BAG bag44.htm bag442 2 ref harv refend Algebraic curves navbox Category Riemann ... more details
, where g 2 d &minus f &minus m 2 is the Genus mathematics genus of math S math . The intersection ... sup math Q math arises as the Seifert matrix of a knot with genus g Seifert surface. The Alexander ... of a knot Seifert surfaces are not at all unique a Seifert surface S of genus g and Seifert matrix V can be modified by a surgery theory surgery , to be replaced by a Seifert surface S of genus g ... K is the knot invariant defined by the minimal genus mathematics genus g of a Seifert surface for K ... 1 q &minus 1 2 The degree of the Alexander polynomial is a lower bound on twice the genus of the knot ...File Borromean Seifert surface.png thumb A Seifert surface bounded by a set of Borromean rings . In mathematics , a Seifert surface named after Germany German mathematician Herbert Seifert ref Cite journal ... 12 issue 4 pages 485 496 year 2006 doi 10.1109 TVCG.2006.83 ref is a surface whose boundary of a manifold ... easily calculated using a Seifert surface. Seifert surfaces are also interesting in their own right ... in Euclidean space Euclidean 3 space or in the 3 sphere . A Seifert surface is a compact space compact , connected space connected , oriented surface S embedded in 3 space whose boundary is L such that the orientation ... of S has non empty boundary. Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3 space is the Seifert surface associated to its boundary link. A single knot or link can have many different inequivalent Seifert surfaces. A Seifert surface must be oriented . It is possible ... File Moebiusband wikipedia.png 150px right thumb A Seifert surface. Unlike a M bius strip it has two ... to be a Seifert surface for the unknot because it is not orientable. The checkerboard ... half twists. As with the previous example, this is not a Seifert surface as it is not orientable. Applying Seifert s algorithm to this diagram, as expected, does produce a Seifert surface in this case, it is a punctured torus of genus g 1 , and the Seifert matrix is math V begin pmatrix 1 & 1 0 & 1 ... more details
In mathematics , an algebraic surface is an algebraic variety of dimension of an algebraic variety dimension two. In the case of geometry over the field of complex number s, an algebraic surface has complex dimension two as a complex manifold , when it is non singular and so of dimension four as a smooth manifold . The theory of algebraic surfaces is much more complicated than that of algebraic curve s including the compact space compact Riemann surface s, which are genuine surface s of real dimension two . Many results were obtained, however, in the Italian school of algebraic geometry , and are up to 100 years old. Examples of algebraic surfaces include is the Kodaira dimension &minus the complex projective plane projective plane , quadric s in P sup 3 sup , cubic surface s, Veronese surface , del Pezzo surface s, ruled surface s 0 K3 surface s, abelian surface s, Enriques surface s, hyperelliptic surface s 1 Elliptic surface s 2surface of general type surfaces of general type . For more examples see the list of algebraic surfaces . The first five examples are in fact birationally equivalent . That is, for example, a cubic surface has a function field of an algebraic variety function field isomorphic to that of the projective plane , being the rational function s in two indeterminates. The cartesian product of two curves also provides examples. The birational geometry ... . The general type class, of Kodaira dimension 2, is very large degree 5 or larger for a non singular surface in P sup 3 sup lies in it, for example . There are essential three Hodge number invariants of a surface. Of those, h sup 1,0 sup was classically called the irregularity and denoted by q and h sup 2,0 sup was called the geometric genus p sub g sub . The third, h sup 1,1 sup , is not a birational ... genus p sub a sub is the difference geometric genus &minus irregularity. In fact this explains ... http maxwelldemon.com 2009 03 29 surfaces 2 algebraic surfaces Overview and thoughts on designing ... more details
and are elliptic surface s of genus 0. They are quotients of K3 surface s by a group of order 2 acting without fixed points and their theory is similar to that of algebraic K3 surfaces. Enriques surfaces can also be defined over other fields. Over fields of characteristic other than 2, Michael ... sup sup is Z 2Z. The surface is a quotient of a reduced singular Gorenstein surface by the group scheme sub 2 sub . Singular dim H sup 1 sup O 1 and is acted on non trivially by the Frobenius endomorphism. This implies K 0, and Pic sup sup is sub 2 sub . The surface is a quotient of a K3 surface by the group scheme Z 2Z. Supersingular dim H sup 1 sup O 1 and is acted on trivially by the Frobenius endomorphism. This implies K 0, and Pic sup sup is sub 2 sub . The surface is a quotient of a reduced singular Gorenstein surface by the group scheme sub 2 sub . All Enriques surfaces ... general homogeneous polynomial Q of degree 2. Then its normalization is an Enriques surface. This is the original ... free involution is an Enriques surface, and all Enriques surfaces in characteristic other than 2 ... year 1949 , are complex algebraic surface s such that the irregularity q 0 and the canonical line ... of characteristic 2 the definition is modified, and there are two new families, called singular and supersingular ... plurigenera P sub n sub are 1 if n is even and 0 if n is odd. The fundamental group has order 2. The second cohomology group H sup 2 sup X , Z is isomorphic to the sum of the unique even unimodular lattice II sub 1,9 sub of dimension 10 and signature 8 and a group of order 2. Hodge diamond table border 0 cellpadding 2 cellspacing 0 tr th th th th th 1 th tr tr th th th 0 th th th th 0 th tr tr .... Characteristic 2 In characteristic 2 there are some new families of Enriques surfaces, sometimes ... . In characteristic 2 the definition of Enriques surfaces is modified they are defined to be minimal ... is 10. In characteristics other than 2 this is equivalent to the usual definition. There are now ... more details
for the algebraic surface Klein icosahedral surface In mathematics, a Klein surface , named after Felix Klein , is a non orientable closed surface . A Klein surface is homeomorphism homeomorphic to a connected sum of a number of copies of the real projective plane . The case of two copies corresponds to the famous Klein bottle . The next case, the connected sum of three copies of RP sup 2 sup , is a surface of Euler characteristic equal to &minus 1. Its orientable double cover is the genus2surface . Category Surfaces topology stub ... more details
For the term in index theory Genus of a multiplicative sequence Refimprove date April 2010 In mathematics , genus plural genera has a few different, but closely related, meanings Topology Orientable surface This section is linked from Complex plane The genus of a connected, orientability orientable surface ... , via the relationship 22 g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads 22 g b . For instance The sphere S sup 2 sup and a disk mathematics disc both have genus zero. Image Mug and Torus morph.gif 100px right Donut or coffee cup? A torus has genus one, as does the surface of a coffee ... apart from their coffee mug. An explicit construction of surfaces of genus g is given in the article on the fundamental polygon . gallery caption Genus of orientable surfaces widths 100px heights 100px perrow 6 File Sphere filled blue.svg genus 0 File Torus illustration.png genus 1 File Double torus illustration.png genus2 File Triple torus illustration.png genus 3 gallery In simpler terms, the value of an orientable surface s genus is equal to the number of holes it has. ref http mathworld.wolfram.com Genus.html ref Non orientable surfaces The orientability non orientable genus , demigenus , or Euler genus of a connected, non orientable closed surface is a positive integer representing ... surface in terms of the Euler characteristic , via the relationship 2 k , where k is the non orientable ... genus of all Seifert surface s for K . A Seifert surface of a knot is however a manifold with boundary the boundary being the knot, i.e. homeomorphic to the unit circle. The genus of such a surface ... genus zero. A solid torus math D 2 times S 1 math has genus one. Graph theory Main Graph embedding ... crossing itself on a sphere with n handles i.e. an oriented surface of genus n . Thus, a planar ... crossing itself on a sphere with n cross caps i.e. a non orientable surface of non orientable genus ... more details
2t , 0 r t & cos t cos 2 t , cos t sin 2 t, sin t end align math one obtains a ruled surface that contains the M bius strip . Alternatively, a ruled surface can be parametric model parametrized as math ... of a 2 dimensional vector bundle over some curve. The ruled surfaces with base curve of genus 0 ...Image Ruled hyperboloid.jpg thumb right A hyperboloid of one sheet is a doubly ruled surface it can be generated by either of two families of straight lines. In geometry , a surface S is ruled also called ... examples are the plane mathematics plane and the curved surface of a cylinder geometry cylinder or cone geometry cone . Other examples are a conical surface with ellipse elliptical directrix , the right conoid , the helicoid , and the tangent developable of a smooth curve in space. A ruled surface ... . A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains three distinct lines through each of its points ... means that a ruled surface has a parametric equation parametric representation of the form math S t,u p t u r t math where math S t,u math is the generic point on the surface, math p t math is point that traces a curve lying on the surface, and math r t math is a unit vector unit length vector that traces ... on the surface. In particular, when math p t math and math q t math move with constant speed along two skew lines , the surface is a hyperbolic paraboloid , or a piece of an hyperboloid of one sheet. Developable surface A developable surface is a surface that can be locally unrolled onto a flat plane without tearing or stretching it. If a developable surface lies in three dimensional Euclidean ... is not. More generally, any developable surface in three dimensions is part of a complete ruled surface ... line on the surface through any given point, and this condition is now often used as the definition ... more details
, Ann. Math. Studies 79 1974 207 226 ref For genus2 the order is maximized by the Bolza surface ... curve given by an equation y sup 2 sup x sup 3 sup a x b . Tori are the only Riemann surfaces of genus mathematics genus one, surfaces of higher genera g are provided by the hyperelliptic surface s y ... sqrt.jpg math f z z frac 1 2 math File Riemann surface cube root.jpg math f z z frac 1 3 math File ... of genus g , 3 g &minus 3 complex parameters suffice. When a hyperbolic surface is compact, then the total area of the surface is 4 g &minus 1 , where g is the genus mathematics genus of the surface the area is obtained by applying the Gauss Bonnet theorem to the area ... lattice have addition symmetries from rotation by 90 and 60 . For genus2, the isometry group ... non abelian simple group. For genus 4, Bring s surface is a highly symmetric surface. For genus 7 the order ...for the Riemann surface of a subring of a field Zariski Riemann space Image Riemann sqrt.jpg thumb right Riemann surface for the function &fnof z &radic z . The two horizontal axes represent ... in complex analysis , a Riemann surface , first studied by and named after Bernhard Riemann ... and other algebraic function s, or the natural logarithm logarithm . Every Riemann surface is a two dimensional real analytic manifold i.e., a surface , but it contains more structure specifically a complex ... real manifold can be turned into a Riemann surface usually in several inequivalent ways if and only .... Definitions There are several equivalent definitions of a Riemann surface. A Riemann surface X is a complex ... of the complex plane to the Riemann surface is called a chart . Additionally, the transition map s between two overlapping charts are required to be Holomorphic function holomorphic . A Riemann surface is an oriented manifold oriented manifold of real dimension two a two sided surface together with a conformal ... 150px The Riemann sphere. The complex plane C is the most basic Riemann surface. The map f z z the identity ... more details
In Riemann surface theory and hyperbolic geometry , the Macbeath surface , also called Macbeath s curve or the Fricke Macbeath curve , is the genus 7 Hurwitz surface . The automorphism group of the Macbeath surface is the simple group Projective linear group PSL 2,8 , consisting of 504 symmetries. ref name w harvtxt Wohlfahrt 1985 . ref Triangle group construction The surface s Fuchsian group can be constructed as the principal congruence subgroup of the 2,3,7 triangle group in a suitable tower of principal congruence subgroups. Here the choices of quaternion algebra and Hurwitz quaternion order are described at the triangle group page. Choosing the ideal math langle 2 rangle math in the ring of integers, the corresponding principal congruence subgroup defines this surface of genus 7. Its systolic geometry systole is about 5.796, and the number of systolic loops is 126 according to R. Vogeler s calculations. Historical note This surface was originally discovered by harvs first Robert last Fricke authorlink Robert Fricke year 1899 txt , but named after Alexander Macbeath Alexander Murray Macbeath due to his later independent rediscovery of the same curve. ref harvtxt Macbeath 1965 . ref Elkies writes that the equivalence between the curves studied by Fricke and Macbeath may first have been observed by Jean Pierre Serre Serre in a 24.vii.1990 letter to Shreeram Shankar Abhyankar Abhyankar ... matrix of Macbeath s curve of genus seven title Curves, Jacobians, and abelian varieties, Amherst ... Study of the symmetries of the Macbeath surface title Mathematical contributions pages 375 385 publisher ... Mathematische Annalen volume 52 year 1899 pages 321 339 doi 10.1007 BF01476163 issue 2 3 . citation ... On a curve of genus 7 journal Proceedings of the London Mathematical Society volume 15 year 1965 pages ... 247 mr 0819842 doi 10.1017 S0017089500006212 . Corrigendum, vol. 28, no. 2, 1986, p. 241, MR ... Category Differential geometry of surfaces Category Systolic geometry fr Surface de Macbeath sl Macbeathova ... more details
In mathematics, an abelian surface is 2 dimensional abelian variety . One dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic. The algebraic ones are called abelian surfaces and are exactly the 2 dimensional abelian variety abelian varieties . Most of their theory is a special case of the theory of higher dimensional tori or abelian varieties. Criteria to be a product of two elliptic curves up to isogeny were a popular study in the nineteenth century. Invariants The plurigenera are all 1. The surface is diffeomorphic to S sup 1 sup × S sup 1 sup × S sup 1 sup × S sup 1 sup so the fundamental group is Z sup 4 sup . Hodge diamond table border 0 cellpadding 2 cellspacing 0 tr th th th th th 1 th tr tr th th th 2 th th th th 2 th tr tr th 1 th th th th 4 th th th th 1 th tr tr th th th 2 th th th th 2 th tr tr th th th th th 1 th tr table Examples A product of two elliptic curves. The Jacobian variety of a genus2 curve. References Citation last1 Barth first1 Wolf P. last2 Hulek first2 Klaus last3 Peters first3 Chris A.M. last4 Van de Ven first4 Antonius title Compact Complex Surfaces publisher Springer Verlag, Berlin series Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. isbn 978 3 540 00832 3 mr 2030225 year 2004 volume 4 Citation last1 Beauville first1 Arnaud title Complex algebraic surfaces publisher Cambridge University Press edition 2nd series London Mathematical Society Student Texts isbn 978 0 521 49510 3 978 0 521 49842 5 mr 1406314 year 1996 volume 34 eom id a a110040 first Ch. last Birkenhake DEFAULTSORT Abelian Surface Category Algebraic surfaces Category Complex surfaces math stub ... more details
along a non singular degree 6 curve is a genus2 K3 surface. A Kummer surface is the quotient of a two dimensional abelian variety A by the action a &minus a . This results in 16 singularities, at the 2 torsion points of A . The Resolution of singularities minimal resolution of this quotient is a genus 3 K3 surface. A non singular degree 4 surface in P sup 3 sup is a genus 3 K3 surface. The intersection of a quadric and a cubic in P sup 4 sup gives genus 4 K3 surfaces. The intersection of three quadric s in P sup 5 sup gives genus 5 K3 surfaces. harvtxt Brown 2007 describes a computer database of K3 surfaces. See also Supersingular K3 surface Classification of algebraic surfaces References ... for the name K3 surface thumb right width 30 In mathematics , a K3 surface is a complex or algebraic smooth minimal complete surface that is irregularity of a surface regular and has trivial canonical ... be embedded in any projective space as a surface defined by polynomial equations. Andr harvtxt Weil ... that can be used to characterize a K3 surface . The only complete smooth surfaces with trivial ... the latter to define K3 surfaces. Over the complex numbers the condition that the surface is simply ..., 0, 22, 0, 1. The Hodge diamond is table border 0 cellpadding 2 cellspacing 0 tr th th th th th 1 ... theorem for complex K3 surfaces. If M is the set of pairs consisting of a complex K3 surface S and a K hler ... , d &minus 2, , d 0. Projective K3 surfaces If L is a line bundle on a K3 surfaces, then the curves in the linear system have genus g where c sub 1 sub sup 2 sup L 2 g 2. A K3 surface with a line bundle L like this is called a K3 surface of genus g . A K3 surface may have many different line bundles making it into a K3 surface of genus g for many different values of g . The space of sections of the line bundle has dimension g 1, so there is a morphism of the K3 surface to projective space of dimension ... L with c sub 1 sub sup 2 sup L 2 g 2, which is nonempty of dimension 19 for g 2. harvtxt Mukai 2006 ... more details
In mathematics, a Burniat surface is one of the surfaces of general type introduced by harvtxt Burniat 1966 . Invariants The geometric genus and irregularity are both 0. The Chern number c sub 1 sub sup 2 sup is 2, 3, 4, 5, or 6. References Citation last1 Barth first1 Wolf P. last2 Hulek first2 Klaus last3 Peters first3 Chris A.M. last4 Van de Ven first4 Antonius title Compact Complex Surfaces publisher Springer Verlag, Berlin series Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. isbn 978 3 540 00832 3 mr 2030225 year 2004 volume 4 Citation last1 Burniat first1 Pol title Sur les surfaces de genre P sub 12 sub 1 doi 10.1007 BF02413731 mr 0206810 year 1966 journal Annali di Matematica Pura ed Applicata. Serie Quarta issn 0003 4622 volume 71 issue 1 pages 1 24 Category Algebraic surfaces Category Complex surfaces ... more details
Kummer quartic surface directly from the Jacobian of a genus2 curve The Jacobian of math C math maps ... of genus2, we may identify the Jacobian math Jac C math with math Pic 2 C math under the map math ... theta characteristics in this in genus2 . Hence the branch points of the canonical map math ... of curves of genus two, every non trivial 2 torsion point is uniquely expressed as a difference ... surface , first studied by harvs txt authorlink Ernst Kummer last Kummer year 1864 , is an irreducible algebraic surface of degree 4 in math mathbb P 3 math with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian of a smooth hyperelliptic curve of genus mathematics genus2 i.e. a quotient of the Jacobian by the Kummer involution x &minus 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 surface closely related to Kummer surfaces include Weddle surface s, Wave surface s, and tetrahedroid s. Geometry of the Kummer surface Singular quartic surfaces and the double plane model Let math K subset mathbb P 3 math be a quartic surface, and let p be a singular point of this surface. Identifying the lines in math mathbb P 3 math through the point p with math mathbb P 2 math , we get a double cover from the blow up of K at p to math mathbb P 2 math this double cover is given by sending q p ... map to nodes of C . By the genus degree formula , the maximal number possible number of nodes ... from the symmetric product math Sym 2 C math to math Pic 2 C math , defined by math p,q mapsto p q math ... C to K C math is a double cover. Hence we get a double cover math Kum C to Sym 2 K C math . This double ... more details
charges near the surface of a 2 dimensional crystal with charge election density around sphere ... 2. In this case, stress of one surface is changed upon deposition of material which results the bending ... materials Nanowire materials Nanoporous materials References Reflist 2 DEFAULTSORT Surface ...Surface stress was first defined by Josiah Willard Gibbs ref J.W. Gibbs, The Scientific Papers of J. Willard ... area needed to elastically stretch a pre existing surface . A similar term called surface free energy ... a new surface, is easily confused with surface stress . Although surface stress and surface free ... terms represent a force per unit length , they have been referred to as surface tension , which contributes further to the confusion in the literature. Thermodynamics of surface stress Definition of surface ... math dA math of surface, expressed as math dw gamma dA math Gibbs was the first to define another surface ... area needed to elastically stretch a pre existing surface. Surface stress can be derived from surface ... One can define a surface stress tensor math f ij math that relates the work associated with the variation in math gamma A math , the total excess free energy of the surface, owing to the strain math ... are elastically strained. The work associated with the first step unstrained is math W 1 2 gamma ... of new surfaces. For the second step, work math w 2 math , equals the work needed to elastically deform ... surfaces. The difference math w 2 w 1 math is equal to the excess work needed to elastically deform two surfaces of area math A 0 math to area math A e ij math or math w 2 w 1 2 int f ij d A epsilon ij 2 int Af ij d epsilon ij math the work associated with the second step of the second path can be expressed as math W 22 gamma e ij A e ij math , so that math W 2 W 1 2 gamma e ij A e ij gamma 0 A 0 math These two paths are completely reversible, or W sub 2 sub W sub 1 sub w sub 2 sub W sub 1 sub . It means math 2 gamma epsilon ij A epsilon ij gamma 0 A 0 2 int Af ij d epsilon ij math Since ... more details
A cubic surface is a projective variety studied in algebraic geometry . It is an algebraic surface in three ... of degree 3 hence, cubic . Cubic surfaces are del Pezzo surface s. Examples If math mathbb ... 3 Z 3 W 3 0 math is a cubic surface called the Fermat cubic surface . The Clebsch surface is the set of points where math X 3 Y 3 Z 3 W 3 X Y Z W 3 math Cayley s nodal cubic surface is the set of points where math WXY XYZ YZW ZWX 0 math 27 lines on a cubic surface The Cayley Salmon theorem states that a smooth variety smooth cubic surface over an algebraically closed field contains 27 straight lines ... lines with self intersection number &minus 1, or in other words the &minus 1 curves on the surface. An Eckardt point is a point where 3 of the 27 lines meet. A smooth cubic surface can also be described as a rational surface obtained by blowing up six points in the projective plane in general position ... lines in math mathbb P 2 math which join two of the blown up points, and the proper transforms of the 6 conics in math mathbb P 2 math which contain all but one of the blown up points. Clebsch gave a model of a cubic surface, called the Clebsch diagonal surface , where all the 27 lines are defined ... acts as the U duality group. This map between del Pezzo surface s and M theory on tori is known ... planes of a canonic sextic curve of genus 4, form a ADE classification Trinities trinity in the sense ... is Cayley s nodal cubic surface math WXY XYZ YZW ZWX 0 math with 4 nodal Mathematical singularity ... issn 0024 6107 volume 19 issue 2 pages 245 256 Citation last1 Cayley first1 Arthur author1 link ... upon the cubic surface publisher Merchant books series Reprinting of Cambridge Tracts in Mathematics ... Berlin, New York series Lecture Notes in Mathematics isbn 978 3 540 61795 2 doi 10.1007 BFb0094399 ... title Cubic surface urlname CubicSurface http demonstrations.wolfram.com LinesOnACubicSurface Lines on a Cubic Surface by Ryan Hoban The Experimental Geometry Lab at the University of Maryland based ... more details
File Endrass surface.png thumb 300px right in algebraic geometry, an Endrass surface is a surface of degree 8 with 168 nodes found by harvs txt last Endrass year 1997 . See also Barth surface Sarti surface Togliatti surface References Citation last1 Endrass first1 S. title A projective surface of degree eight with 168 nodes url http arxiv.org abs alg geom 9507011 id MR 1489118 year 1997 journal Journal of Algebraic Geometry issn 1056 3911 volume 6 issue 2 pages 325 334 Category Algebraic surfaces ... more details
In algebraic geometry , a White surface is one of the rational surface s in P sup n sup studied by harvtxt White 1923 , generalizing cubic surface s and Bordiga surface s, which are the cases n 3 or 4. A White surface in P sup n sup is given by the embedding of P sup 2 sup blowing up blown up in n n 1 2 points by the linear system of degree n curves through these points. References citation first F. P. last White title On certain nets of plane curves journal Proceedings of the Cambridge philosophical society volume 22 year 1923 pages 1&ndash 10 doi 10.1017 S0305004100000037 Category complex surfaces Category algebraic surfaces ... more details
In mathematics, a Catanese surface is one of the surfaces of general type introduced by harvtxt Catanese 1981 . Construction The construction starts with a quintic V with 20 double points. Let W be the surface obtained by blowing up the 20 double points. Suppose that W has a double cover X branched over the 20 exceptional &minus 2 curves. Let Y be obtained from X by blowing down the 20 &minus 1 curves in X . If there is a group of order 5 acting freely on all these surfaces, then the quotient Z of Y by this group of order 5 is a Catanese surface. Catanese found a 4 dimensional family of curves constructed like this. Invariants The fundamental group is trivial. The irregularity and the geometric genus are both 0. Hodge diamond table border 0 cellpadding 2 cellspacing 0 tr th th th th th 1 th tr tr th th th 0 th th th th 0 th tr tr th 0 th th th th 8 th th th th 0 th tr tr th th th 0 th th th th 0 th tr tr th th th th th 1 th tr table References Citation last1 Barth first1 Wolf P. last2 Hulek first2 Klaus last3 Peters first3 Chris A.M. last4 Van de Ven first4 Antonius title Compact Complex Surfaces publisher Springer Verlag, Berlin series Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. isbn 978 3 540 00832 3 mr 2030225 year 2004 volume 4 Citation last1 Catanese first1 F. title Babbage s conjecture, contact of surfaces, symmetric determinantal varieties and applications doi 10.1007 BF01389064 mr 620679 year 1981 journal Inventiones Mathematicae issn 0020 9910 volume 63 issue 3 pages 433 465 Category Algebraic surfaces Category Complex surfaces ... more details
Unreferenced date November 2009 In mathematics , in the fields of differential geometry and algebraic geometry , the Enneper surface is a surface that can be described parametrically by math x u 1 u 2 3 v 2 3, math math y v 1 v 2 3 u 2 3, math math z u 2 v 2 3. math It was introduced by Alfred Enneper in connection with minimal surface theory. Image EnneperSurface.PNG Figure 1. An Enneper surface Implicitization methods of algebraic geometry can be used to find out that the points in the Enneper surface given above satisfy the degree 9 polynomial equation math 64 z 9 128 z 7 64 z 5 702 x 2 y 2 z 3 18 x 2 y 2 z 144 y 2 z 6 x 2 z 6 math math 162 y 4 z 2 x 4 z 2 27 y 6 x 6 9 x 4 z y 4 z 48 x 2 z 3 y 2 z 3 math math 432 x 2 z 5 y 2 z 5 81 x 4 y 2 x 2 y 4 240 y 2 z 4 x 2 z 4 135 x 4 z 3 y 4 z 3 0. math Image EnneperSurface2.PNG Figure 2. The Enneper surface in Figure 1 has been rotated 30 around the z axis. Image EnneperSurface3.PNG Figure 3. The Enneper surface in Figure 1 has been rotated 60 around the z axis. Dually, the tangent plane at the point with given parameters is math a b x c y d z 0, math where math a u 2 v 2 1 u 2 3 v 2 3 , math math b 6 u, math math c 6 v, math math d 3 1 u 2 v 2 . math Its coefficients satisfy the implicit degree 6 polynomial equation math 162 a 2 b 2 c 2 6 b 2 c 2 d 2 4 b 6 c 6 54 a b 4 d a c 4 d 81 a 2 b 4 a 2 c 4 math math 4 b 4 c 2 b 2 c 4 3 b 4 d 2 c 4 d 2 36 a b 2 d 3 a c 2 d 3 0. math Enneper s is a minimal surface . The Jacobian , Gaussian curvature and mean curvature are math J 1 u 2 v 2 4 81, math math K 4 9 J, math math H 0. math References MathWorld title Enneper s Minimal Surface urlname EnnepersMinimalSurface DEFAULTSORT Enneper Surface Category Algebraic surfaces Category Algebraic geometry Category Minimal surfaces fr Surface d Enneper it Superficie di Enneper nl Enneper oppervlak sl Enneperjeva ploskev ... more details
acting on C sup 2 sup by multiplication by 2 this was the first example of a compact complex surface ...In complex geometry , a Hopf surface is a compact complex surface obtained as a quotient of the complex vector space with zero deleted C sup 2 sup 0 by a Group action free action of a discrete group. If this group is the integers the Hopf surface is called primary , otherwise it is called secondary . Some authors use the term Hopf surface to mean primary Hopf surface . The first example ... &infin and all their plurigenera vanish. The geometric genus is 0. The fundamental group has a normal ... 2 cellspacing 0 tr th th th th th 1 th tr tr th th th 0 th th th th 1 th tr tr th 0 th th th th ... that that a compact complex surface with vanishing the second Betti number and whose fundamental group contains an infinite cyclic subgroup of finite index is a Hopf surface. Primary Hopf surfaces ... Kodaira Kodaira classified the primary Hopf surfaces. A primary Hopf surface is obtained as math H bigg Bbb C 2 backslash 0 bigg Gamma, math where math Gamma math is a group generated by a polynomial ... 0, the Hopf surface is an elliptic fiber space over the projective line if &alpha sup m sup &beta ... of any primary Hopf surface is isomorphic to the non zero complex numbers C sup sup . harvtxt Kodaira 1966b has proven that a complex surface is diffeomorphic to S sup 3 sup × S sup 1 sup if and only if it is a primary Hopf surface. Secondary Hopf surfaces Any secondary Hopf surface has a finite unramified cover that is a primary Hopf surface. Equivalently, its fundamental group has a subgroup ... jmath1948&cdvol 27&noissue 2&startpage 222 id MathSciNet id 0402128 year 1975 journal Journal of the Mathematical ... 2 Citation last1 Kato first1 Masahide title Erratum to Topology of Hopf surfaces url http www.journalarchive.jst.go.jp ... title Complex structures on S sup 1 sup S sup 3 sup url http www.pnas.org content 55 2 240.full.pdf ... 55 pages 240 243 doi 10.1073 pnas.55.2.240 issue 2 Citation last1 Matumoto first1 Takao last2 ... more details
circular conical surface of aperture math 2 theta math , whose axis is the math z math coordinate axis ..., y and z, the general equation for a cone with apex at origin is a homogenous equation of degree 2 given by math S x, y, z ax 2 by 2 cz 2 2uxy 2vyz 2wzx 0 math See also Conic section Developable surface ...Unreferenced date December 2009 Image DoubleCone.png thumb 250px right A circular conical surface In geometry , a general conical surface is the unbounded surface formed by the union of all the straight ... lines is called a generatrix of the surface. Every conic surface is ruled surface ruled and developable surface developable . In general, a conical surface consists of two congruent unbounded halves ..., 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 math C math , and the apex ... to its plane , one obtains the right circular conical surface . This special case is often called ... math 2 theta math . More generally, when the directrix math C math is an ellipse , or any ... quadric , which is a special case of a quadric quadric surface . A cylindrical surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction. Indeed, in projective geometry a cylindrical surface is just a special case of a conical surface. Equations A conical surface math S math can be described parametrization parametrically as math ... infty, infty math , respectively. In implicit geometric model implicit form, the same surface is described by math S x,y,z 0 math where math S x,y,z x 2 y 2 cos theta 2 z 2 sin theta 2. math More generally, a right circular conical surface with apex at the origin, axis parallel to the vector math d math , and aperture math 2 theta math , is given by the implicit vector calculus vector equation math S u 0 math where math S u u . d 2 d . d u . u cos theta 2 math or math S u u . d d u cos theta ... more details
Encyclopedia
top
