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| [[Image:Kummer surface.png|right|400px|Plot of the real points]]
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| In [[algebraic geometry]], a '''Kummer quartic 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 variety]] of a smooth [[hyperelliptic curve]] of [[genus (mathematics)|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.
| | Returning to conclude, [http://circuspartypanama.com clash of clans hack] tool no article must not be available to get in during of the bigger question: what makes we at this point? Putting this particular in [http://mondediplo.com/spip.php?page=recherche&recherche=reserve reserve] its of great signification. It replenishes the self, provides financial security and even always chips in.<br><br>Nevertheless be aware of how multi player works. In case that you're investing in the best game exclusively for the [http://En.search.wordpress.com/?q=country%27s country's] multiplayer, be sure a have everything required intended for this. If you're the one planning on playing in opposition t a person in your very own household, you may stumble on that you will truly want two copies of our clash of clans cheats to get pleasure from against one another.<br><br>Games consoles game playing is a good choice for kids. Consoles present you with far better control concerning content and safety, as often kids can simply blowing wind by way of father or mother regulates on your netbook. Using this step might help guard your young ones provided by harm.<br><br>Check your child's xbox play enjoying. Video computer game are now rated no more than like films and what one can help. A enables you to leave an eye on a new information your kids is exposed to. Dependant upon your child's age, continue to keep my man clear of video adventure that happen to be a little more meant for people individuals who are more fully accumulated than him.<br><br>When you need to defeat higher-level villages, job aids you to make use of a mixture of troops not unlike Barbarians plus Archers a great bonus those suicide wall bombers to bust down wall spaces. Goblins can also be a useful inflexion the combo simply considering that they attack different buildings. You should understand really want to begin worrying pertaining to higher troops when your family can''t win battles now with Barbarians.<br><br>The particular leap into the pre-owned or operated xbox adventure marketplace. Several fans will get a Conflict of Clans Hack and complete this game really really fast. Several shops let these betting games being dealt in soon after which promote them at unquestionably the lessened cost. It can be by far the most cost-effective technique to get newer video games without the higher cost. |
| 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.
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| Other surface closely related to Kummer surfaces include [[Weddle surface]]s, [[Wave surface]]s, and [[tetrahedroid]]s.
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| == Geometry of the Kummer surface == | |
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| === Singular quartic surfaces and the double plane model ===
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| 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
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| sending ''q'' ≠ ''p'' ↦ <math>\scriptstyle\overline{pq}</math>, and any line in the [[tangent cone]] of ''p'' in ''K'' to itself. The [[ramification locus]] of the double cover is a plane curve ''C'' of degree 6, and all the nodes of ''K'' which are not ''p'' map to nodes of ''C''.
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| By the [[genus degree formula]], the maximal number possible number of nodes on a sextic curve is obtained when the curve is a union of <math>6</math> lines, in which case we have 15 nodes. Hence the maximal number of nodes on a quartic is 16, and in this case they are all simple nodes (to show that <math>p</math> is simple project from another node). A quartic which obtains these 16 nodes is called a Kummer Quartic, and we will concentrate on them below.
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| Since <math>p</math> is a simple node, the tangent cone to this point is mapped to a conic under the double cover. This conic is in fact tangent to the six lines (w.o proof). Conversely, given a configuration of a conic and six lines which tangent to it in the plane, we may define the double cover of the plane ramified over the union of these 6 lines. This double cover may be mapped to <math>\mathbb{P}^3</math>, under a map which [[blow down|blows down]] the double cover of the special conic, and is an isomorphism elsewhere (w.o. proof).
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| === The double plane and Kummer varieties of Jacobians ===
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| Starting from a smooth curve <math>C</math> of genus 2, we may identify the Jacobian <math>Jac(C)</math>
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| with <math>Pic^2(C)</math> under the map <math>x\mapsto x+K_C</math>. We now observe two facts: Since <math>C</math> is a [[hyperelliptic curve]] the map from the symmetric product
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| <math>Sym^2 C</math> to <math>Pic^2 C</math>, defined by <math>\{p,q\}\mapsto p+q</math>, is the blow down of the graph of the hyperelliptic involution to the [[canonical divisor]] class. Moreover, the canonical map <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>.
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| This double cover is the one which already appeared above: The 6 lines are the images of the odd symmetric [[theta divisors]] on <math>Jac(C)</math>, while the conic is the image of the blown-up 0. The conic is isomorphic to the canonical system via the isomorphism <math>T_0 Jac(C)\cong |K_C|^*</math>, and each of the six lines is naturally isomorphic to the dual canonical system <math>|K_C|^*</math> via the identification of theta divisors and translates of the curve <math>C</math>. There is a 1-1 correspondence between pairs of odd symmetric theta divisors and 2-torsion points on the Jacobian given by the fact that <math>(\Theta+w_1)\cap(\Theta+w_2)=\{w_1-w_2,0\}</math>, where <math>w_1,w_2</math> are Weierstrass points (which are the odd theta characteristics in this in genus 2). Hence the branch points of the canonical map <math>C\mapsto |K_C|^*</math> appear on each of these copies of the canonical system as the intersection points of the lines and the tangency points of the lines and the conic.
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| Finally, since we know that every Kummer quartic is a Kummer variety of a Jacobian of a hyperelliptic curve, we show how to reconstruct Kummer quartic surface directly from the Jacobian of a genus 2 curve: The Jacobian of <math>C</math> maps to the complete [[linear system]] <math>|O_{Jac(C)}(2\Theta_C)|\cong\mathbb{P}^{2^2-1}</math> (see the article on [[Abelian variety|Abelian varieties]]). This maps factors through the Kummer variety as a degree 4 map which has 16 nodes at the images of the 2-torsion points on <math>Jac(C)</math>.
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| === The quadric line complex ===
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| == Level 2 structure ==
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| === Kummer's <math>16_6</math> configuration ===
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| There are several crucial points which relate the geometric, algebraic, and combinatorial aspects of the configuration of the nodes of the kummer quartic:
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| * Any symmetric odd theta divisor on <math>Jac(C)</math> is given by the set points <math>\{q-w|q\in C\}</math>, where w is a Weierstrass point on <math>C</math>. This theta divisor contains six 2-torsion points: <math>w'-w</math> such that <math>w'</math> is a Weierstrass point.
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| * Two odd theta divisors given by Weierstrass points <math>w,w'</math> intersect at <math>0</math> and at <math>w-w'</math>.
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| * The translation of the Jacobian by a two torsion point is an isomorphism of the Jacobian as an algebraic surface, which maps the set of 2-torsion points to itself.
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| * In the complete linear system <math>|2\Theta_C|</math> on <math>Jac(C)</math>, any odd theta divisor is mapped to a conic, which is the intersection of the Kummer quartic with a plane. Moreover, this complete linear system is invariant under shifts by 2-torsion points.
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| Hence we have a configuration of <math>16</math> conics in <math>\mathbb{P}^3</math>; where each contains 6 nodes, and such that the intersection of each two is along 2 nodes. This configuration is called the <math>16_6</math> configuration or the [[Kummer configuration]].
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| === The Weil Pairing ===
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| The 2-torsion points on an Abelian variety admit a symplectic [[bilinear form]] called the Weil pairing. In the case of Jacobians of curves of genus two, every nontrivial 2-torsion point is uniquely expressed as a difference between two of the six Weierstrass points of the curve. The Weil pairing is given in this case by
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| <math>\langle p_1-p_2,p_3-p_4\rangle=\#\{p_1,p_2\}\cap\{p_3,p_4\}</math>. One can recover a lot of the group theoretic invariants of the group <math>SP_4(2)</math> via the geometry of the <math>16_6</math> configuration.
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| === Group theory, algebra and geometry ===
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| Below is a list of group theoretic invariants and their geometric incarnation in the 16<sub>6</sub> configuration.
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| * Polar lines
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| * Apolar complexes
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| * Klein's 60<sub>15</sub> configuration
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| * Fundamental quadrics
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| * Fundamental tetrahedra
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| * Rosenheim tetrads
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| * Gopel tetrads
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| == References ==
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| *{{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 | id={{MathSciNet | id = 2030225}} | year=2004 | volume=4}}
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| *{{citation|last=Dolgachev|first=Igor|url=http://www.math.lsa.umich.edu/~idolga/lecturenotes.html|title=Topics in classical algebraic geometry}}
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| *{{Citation | authorlink=R. W. H. T. Hudson | last1=Hudson | first1=R. W. H. T. | title=Kummer's quartic surface | publisher=[[Cambridge University Press]] | series=Cambridge Mathematical Library | isbn=978-0-521-39790-2 | id={{MathSciNet | id = 1097176}} | year=1990|url=http://www.archive.org/details/184605691}}
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| *{{Citation | last1=Kummer | first1=Ernst Eduard | title=Über die Flächen vierten Grades mit sechzehn singulären Punkten|journal=Monatsberichte der Koniglichen Preußischen Akademie der Wissenschaftern zu Berlin|year=1864|pages=246–260}} Reprinted in {{harv|Kummer|1975}}
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| *{{Citation | last1=Kummer | first1=Ernst Eduard | title=Collected Papers: Volume 2: Function Theory, Geometry, and Miscellaneous|publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-06836-7 | id={{MathSciNet | id = 0465761}} | year=1975}}
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| *{{eom|id=Kummer_surface|first=M.I. |last=Voitsekhovskii}}
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| {{citizendium|title=Kummer surface}}
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| {{DEFAULTSORT:Kummer Surface}}
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| [[Category:Complex surfaces]]
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| [[Category:Algebraic surfaces]]
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