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| {{about|algebraic geometry|the concept in graph theory|Intersection number (graph theory)}}
| | Your Tribe is the some strong of all as well as have the planet (virtual) at your toes, and all that with only a brief on-line on the web that may direct somebody step by step when how to get all cheat code for Struggle of Tribes.<br><br>[http://Www.Wired.com/search?query=Video+games Video games] are fun to have fun with your kids. This helps you learn much more details about your kid's interests. Sharing interests with children like this can likewise create great conversations. It also gives you an opportunity to monitor developing on their skills.<br><br>Video games are very well-liked numerous homes. The most of people perform online online to pass through time, however, some blessed consumers are paid to experience clash of clans sur pc. Video gaming is going to turn out to be preferred for some work-time into the future. These tips will an individual to if you are going try out online.<br><br>A fantastic method to please your children with a gaming device and ensure they continue to exist fit is to obtain Wii. This the game console . needs real task to play. Your children won't be perched for hours on termination playing clash of clans hack - [http://prometeu.net why not try these out],. They really need to be moving around as the right way to play the games at this particular system.<br><br>Web site not only provides end tools, there is also Clash of Clans chop no survey by anyone. Strict anti ban system hand it over to users to utilize pounds and play without an hindrance. If players are interested in qualifing for the program, they are obviously required to visit great site and obtain our hack tool trainer now. The name of the online site is Amazing Cheats. A number of site have different types from software by which those can get past tough stages in the video game.<br><br>It all construction is what stands that you can be a little more a part of the new clan, however it likewise houses reinforcement troops. Click a button to be ask your clan to send you some troops, and they are starting to be out generally there are to make use off in assaults, or to defend your base for you while you're worries your weekly LARPing company. Upgrading this putting together permits extra troops as a way to be stored for security. You may need to have 20 available slots as a way to get a dragon. This is a top quality base for players seeking to shield trophies in addition to never worried about fontaine. Players will uncover it hard to wash out your city area. Most will take care of for the easy get and take out your own personal assets.<br><br>Which will master game play throughout the shooter video games, grasp your weapons. Discover everything there is realize about each and whatever weapon style in the overall game. Each weapon excels found in certain ways, but belongs short in others. When you know the exact pluses and minuses coming from all each weapon, you can easily use them to registered advantage. |
| In [[mathematics]], and especially in [[algebraic geometry]], the '''intersection number''' generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for [[tangent|tangency]]. One needs a definition of intersection number in order to state results like [[Bézout's theorem]].
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| The intersection number is obvious in certain cases, such as the intersection of ''x''- and ''y''-axes which should be one. The complexity enters when calculating intersections at points of tangency and intersections along positive dimensional sets. For example if a plane is tangent to a surface along a line, the intersection number along the line should be at least two. These questions are discussed systematically in [[intersection theory]].
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| == Definition for Riemann surfaces ==
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| Let ''X'' be a [[Riemann surface]]. Then the intersection number of two closed curves on ''X'' has a simple definition in terms of an integral. For every closed curve ''c'' on ''X'' (i.e., smooth function <math>c : S^1 \to X</math>), we can associate a [[differential form]] <math>\eta_c</math> with the pleasant property that integrals along ''c'' can be calculated by integrals over ''X'':
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| :<math>\int_c \alpha = -\int \int_X \alpha \wedge \eta_c = (\alpha, *\eta_c)</math>, for every closed (1-)differential <math>\alpha</math> on ''X'',
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| where <math>\wedge</math> is the [[wedge product]] of differentials, and <math>*</math> is the [[hodge star]]. Then the intersection number of two closed curves, ''a'' and ''b'', on ''X'' is defined as
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| :<math>a \cdot b := \int \int_X \eta_a \wedge \eta_b = (\eta_a, -*\eta_b) = -\int_b \eta_a</math>.
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| The <math>\eta_c</math> have an intuitive definition as follows. They are a sort of [[dirac delta]] along the curve ''c'', accomplished by taking the differential of a [[unit step function]] that drops from 1 to 0 across ''c''. More formally, we begin by defining for a simple closed curve ''c'' on ''X'', a function ''f<sub>c</sub>'' by letting <math>\Omega</math> be a small strip around ''c'' in the shape of an annulus. Name the left and right parts of <math>\Omega \setminus c</math> as <math>\Omega^{+}</math> and <math>\Omega^{-}</math>. Then take a smaller sub-strip around ''c'', <math>\Omega_0</math>, with left and right parts <math>\Omega_0^{-}</math> and <math>\Omega_0^{+}</math>. Then define ''f<sub>c</sub>'' by
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| :<math>f_c(x) = \begin{cases} 1, & x \in \Omega_0^{-} \\ 0, & x \in X \setminus \Omega^{-} \\ \mbox{smooth interpolation}, & x \in \Omega^{-} \setminus \Omega_0^{-} \end{cases}</math>.
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| The definition is then expanded to arbitrary closed curves. Every closed curve ''c'' on ''X'' is [[Homology (mathematics)|homologous]] to <math>\sum_{i=1}^N k_i c_i</math> for some simple closed curves ''c<sub>i</sub>'', that is,
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| :<math>\int_c \omega = \int_{\sum_i k_i c_i} \omega = \sum_{i=1}^N k_i \int_{c_i} \omega</math>, for every differential <math>\omega</math>. | |
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| Define the <math>\eta_c</math> by
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| :<math>\eta_c = \sum_{i=1}^N k_i \eta_{c_i}</math>.
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| == Definition for algebraic varieties ==
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| The usual constructive definition in the case of algebraic varieties proceeds in steps. The definition given below is for the intersection number of [[Divisor (algebraic geometry)|divisor]]s on a nonsingular variety ''X''.
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| 1. The only intersection number that can be calculated directly from the definition is the intersection of hypersurfaces (subvarieties of ''X'' of codimension one) that are in general position at ''x''. Specifically, assume we have a nonsingular variety ''X'', and ''n'' hypersurfaces ''Z''<sub>''1''</sub>, ..., ''Z''<sub>''n''</sub> which have local equations ''f''<sub>''1''</sub>, ..., ''f''<sub>''n''</sub> near ''x'' for polynomials ''f''<sub>''i''</sub>(''t''<sub>''1''</sub>, ..., ''t''<sub>''n''</sub>), such that the following hold:
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| * <math>n = \dim_k X</math>.
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| * <math>f_i(x) = 0</math> for all ''i''. (i.e., ''x'' is in the intersection of the hypersurfaces.)
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| * <math>\dim_x \cap_{i=1}^n Z_i = 0</math> (i.e., the divisors are in general position.)
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| * The <math>f_i</math> are nonsingular at ''x''.
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| Then the intersection number at the point ''x'' is
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| :<math>(Z_1 \cdots Z_n)_x := \dim_k \mathcal{O}_{X, x} / (f_1, \dots, f_n)</math>,
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| where <math>\mathcal{O}_{X, x}</math> is the local ring of ''X'' at ''x'', and the dimension is dimension as a ''k''-vector space. It can be calculated as the [[Localization of a ring|localization]] <math>k[U]_{\mathfrak{m}_x}</math>, where <math>\mathfrak{m}_x</math> is the maximal ideal of polynomials vanishing at ''x'', and ''U'' is an open affine set containing ''x'' and containing none of the singularities of the ''f''<sub>''i''</sub>.
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| 2. The intersection number of hypersurfaces in general position is then defined as the sum of the intersection numbers at each point of intersection.
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| :<math>(Z_1 \cdots, Z_n) = \sum_{x \in \cap_i Z_i} (Z_1 \cdots Z_n)_x</math>
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| 3. Extend the definition to ''effective'' divisors by linearity, i.e.,
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| :<math>(n Z_1 \cdots Z_n) = n(Z_1 \cdots Z_n)</math> and <math>((Y_1 + Z_1) Z_2 \cdots Z_n) = (Y_1 Z_2 \cdots Z_n) + (Z_1 Z_2 \cdots Z_n)</math>.
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| 4. Extend the definition to arbitrary divisors in general position by noticing every divisor has a unique expression as ''D'' = ''P'' - ''N'' for some effective divisors ''P'' and ''N''. So let ''D''<sub>''i''</sub> = ''P''<sub>''i''</sub> - ''N''<sub>i</sub>, and use rules of the form
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| :<math>((P_1 - N_1) P_2 \cdots P_n) = (P_1 P_2 \cdots P_n) - (N_1 P_2 \cdots P_n)</math>
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| to transform the intersection. | |
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| 5. The intersection number of arbitrary divisors is then defined using a "[[Chow's moving lemma]]" that guarantees we can find linearly equivalent divisors that are in general position, which we can then intersect.
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| Note that the definition of the intersection number does not depend on the order of the divisors.
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| == Further definitions ==
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| The definition can be vastly generalized, for example to intersections along subvarieties instead of just at points, or to arbitrary complete varieties.
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| In algebraic topology, the intersection number appears as the Poincaré dual of the [[cup product]]. Specifically, if two manifolds, ''X'' and ''Y'', intersect transversely in a manifold ''M'', the homology class of the intersection is the [[Poincaré dual]] of the cup product <math>D_M X \smile D_M Y</math> of the Poincaré duals of ''X'' and ''Y''.
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| == Intersection multiplicities for plane curves ==
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| There is a unique function assigning to each triplet (''P'', ''Q'', ''p'') consisting of a pair of polynomials, ''P'' and ''Q'', in ''K''[''x'', ''y''] and a point ''p'' in ''K''<sup>2</sup> a number ''I''<sub>''p''</sub>(''P'', ''Q'') called the ''intersection multiplicity'' of ''P'' and ''Q'' at ''p'' that satisfies the following properties:
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| # <math>I_p(P,Q) = I_p(Q,P).\,</math>
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| # <math>I_p(P,Q)</math> is infinite if and only if ''P'' and ''Q'' have a common factor that is zero at ''p''.
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| # <math>I_p(P,Q)</math> is zero if and only if one of ''P''(''p'') or ''Q''(''p'') is non-zero (i.e. the point p is off one of the curves).
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| # <math>I_p(x,y) = 1</math> where the point ''p'' is at (''x'', ''y'').
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| # <math>I_p(P,Q_1Q_2) = I_p(P,Q_1) + I_p(P,Q_2)</math>
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| # <math>I_p(P + QR,Q) = I_p(P,Q)</math> for any ''R'' in ''K''[''x'', ''y'']
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| Although these properties completely characterize intersection multiplicity, in practice it is realised in several different ways.
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| One realization of intersection multiplicity is through the dimension of a certain quotient space of the power series ring ''K''[<span/>[''x'',''y'']]. By making a change of variables if necessary, we may assume that the point p is (0,0). Let ''P''(''x'', ''y'') and ''Q''(''x'', ''y'') be the polynomials defining the algebraic curves we are interested in. If the original equations are given in homogeneous form, these can be obtained by setting ''z'' = 1. Let ''I'' = (''P'', ''Q'') denote the ideal of ''K''[<span/>[''x'',''y'']] generated by ''P'' and ''Q''. The intersection multiplicity is the dimension of ''K''[<span/>[''x'', ''y'']]/''I'' as a vector space over ''K''.
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| Another realization of intersection multiplicity comes from the [[resultant]] of the two polynomials ''P'' and ''Q''. In coordinates where ''p'' is (0,0), the curves have no other intersections with ''y'' = 0, and the [[degree of a polynomial|degree]] of ''P'' with respect to x is equal to the total degree of ''P'', ''I''<sub>''p''</sub>(''P'', ''Q'') can be defined as the highest power of ''y'' that divides the resultant of ''P'' and ''Q'' (with ''P'' and ''Q'' seen as polynomials over ''K''[''x'']).
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| Intersection multiplicity can also be realised as the number of distinct intersections that exist if the curves are perturbed slightly. More specifically, if ''P'' and ''Q'' define curves which intersect only once in the [[closure (mathematics)|closure]] of an open set U, then for a dense set of (ε,δ) in ''K''<sup>2</sup>, ''P'' − ε and ''Q'' − δ are smooth and intersect transversally (i.e. have different tangent lines) at exactly some number ''n'' points in ''U''. ''I''<sub>''p''</sub>(''P'', ''Q'') = ''n''.
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| ===Example===
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| Consider the intersection of the ''x''-axis with the parabola
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| :<math>y = x^2.\ </math>
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| Then
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| :<math>P = y,\ </math>
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| and
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| :<math>Q = y - x^2,\ </math>
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| so
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| : <math>I_p(P,Q) = I_p(y,y - x^2) = I_p(y,x^2) = I_p(y,x) + I_p(y,x) = 1 + 1 = 2.\,</math>
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| Thus, the intersection degree is two; it is an ordinary [[tangent|tangency]].
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| ==Self-intersections==
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| Some of the most interesting intersection numbers to compute are ''self-intersection numbers''. This should not be taken in a naive sense. What is meant is that, in an equivalence class of [[divisor (algebraic geometry)|divisor]]s of some specific kind, two representatives are intersected that are in [[general position]] with respect to each other. In this way, self-intersection numbers can become well-defined, and even negative.
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| == Applications ==
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| The intersection number is partly motivated by the desire to define intersection to satisfy [[Bézout's theorem]].
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| The intersection number arises in the study [[fixed point (mathematics)|fixed point]]s, which can be cleverly defined as intersections of function [[graph of a function|graph]]s with a [[diagonal]]s. Calculating the intersection numbers at the fixed points counts the fixed points ''with multiplicity'', and leads to the [[Lefschetz fixed point theorem]] in quantitative form.
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| ==References==
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| * {{cite book | author=William Fulton | authorlink=William Fulton (mathematician) | title=Algebraic Curves | series=Mathematics Lecture Note Series | publisher=W.A. Benjamin | year=1974 | isbn=0-8053-3082-8 | pages=74–83 }}
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| * {{cite book | author=Robin Hartshorne | authorlink=Robin Hartshorne | title=Algebraic Geometry | series=[[Graduate Texts in Mathematics]] | volume=52 | year=1977 | isbn=0-387-90244-9 }} Appendix A.
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| * {{cite book | author=William Fulton | authorlink=William Fulton (mathematician) | title=Intersection Theory| series=Mathematics Lecture Note Series | publisher=Springer | year=1998 | isbn= 0-387-98549-2, 9780387985497 }}
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| * ''Algebraic Curves: An Introduction To Algebraic Geometry'', by William Fulton with Richard Weiss. New York: Benjamin, 1969. Reprint ed.: Redwood City, CA, USA: Addison-Wesley, Advanced Book Classics, 1989. ISBN 0-201-51010-3. [http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf Full text online].
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| * {{cite book | author=Hershel M. Farkas | author=Irwin Kra | title=Riemann Surfaces | series=[[Graduate Texts in Mathematics]] | volume=71 | year=1980 | isbn=0-387-90465-4 }}
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| [[Category:Intersection theory]]
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