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Genus–degree Formula
In classical algebraic geometry, the genus–degree formula relates the degree ''d'' of an irreducible plane curve C with its arithmetic genus ''g'' via the formula: :g=\frac12 (d-1)(d-2). Here "plane curve" means that C is a closed curve in the projective plane \mathbb^2. If the curve is non-singular the geometric genus and the arithmetic genus are equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinary singularity of multiplicity ''r'' decreases the genus by \frac12 r(r-1). Proof The proof follows immediately from the adjunction formula. For a classical proof see the book of Arbarello, Cornalba, Griffiths and Harris. Generalization For a non-singular hypersurface H of degree ''d'' in the projective space \mathbb^n of arithmetic genus ''g'' the formula becomes: : g=\binom , \, where \tbinom is the binomial coefficient. Notes See also * Thom conjecture References * * Enrico Arbarello, Maur ... [...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] |
Arithmetic Genus
In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface. Projective varieties Let ''X'' be a projective scheme of dimension ''r'' over a field ''k'', the ''arithmetic genus'' p_a of ''X'' is defined asp_a(X)=(-1)^r (\chi(\mathcal_X)-1).Here \chi(\mathcal_X) is the Euler characteristic of the structure sheaf \mathcal_X. Complex projective manifolds The arithmetic genus of a complex projective manifold of dimension ''n'' can be defined as a combination of Hodge numbers, namely :p_a=\sum_^ (-1)^j h^. When ''n=1'', the formula becomes p_a=h^. According to the Hodge theorem, h^=h^. Consequently h^=h^1(X)/2=g, where ''g'' is the usual (topological) meaning of genus of a surface, so the definitions are compatible. When ''X'' is a compact Kähler manifold, applying ''h''''p'',''q'' = ''h''''q'',''p'' recovers the earlier definition for projective varieties. Kähler manifolds By u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Genus
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds. Definition The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number (equal to by Serre duality), that is, the dimension of the canonical linear system plus one. In other words for a variety of complex dimension it is the number of linearly independent holomorphic -forms to be found on .Danilov & Shokurov (1998), p. 53/ref> This definition, as the dimension of : then carries over to any base field, when is taken to be the sheaf of Kähler differentials and the power is the (top) exterior power, the canonical line bundle. The geometric genus is the first invariant of a sequence of invariants called the plurigenera. Case of curves In the case of complex varieties, (the complex loci of) non-singular curves are Riemann surfaces. The algebraic definition of genus agre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Singularity
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the real function : f(x) = \frac has a singularity at x = 0, where the numerical value of the function approaches \pm\infty so the function is not defined. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. Real analysis In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books by decree in 1586, it is the second oldest university press after Cambridge University Press. It is a department of the University of Oxford and is governed by a group of 15 academics known as the Delegates of the Press, who are appointed by the vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, opposite Somerville College, in the inner suburb of Jericho. For the last 500 years, OUP has primarily focused on the publication of pedagogical texts and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjunction Formula
In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by induction. Adjunction for smooth varieties Formula for a smooth subvariety Let ''X'' be a smooth algebraic variety or smooth complex manifold and ''Y'' be a smooth subvariety of ''X''. Denote the inclusion map by ''i'' and the ideal sheaf of ''Y'' in ''X'' by \mathcal. The conormal exact sequence for ''i'' is :0 \to \mathcal/\mathcal^2 \to i^*\Omega_X \to \Omega_Y \to 0, where Ω denotes a cotangent bundle. The determinant of this exact sequence is a natural isomorphism :\omega_Y = i^*\omega_X \otimes \operatorname(\mathcal/\mathcal^2)^\vee, where \vee denotes the dual of a line bundle. The particular case of a smooth divisor Suppose that ''D'' is a smooth diviso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation :x_1^2+x_2^2+\cdots+x_n^2-1=0 defines an algebraic hypersurface of dimension in the Euclidean space of dimension . This hypersurface is also a smooth manifold, and is called a hypersphere or an -sphere. Smooth hypersurface A hypersurface that is a smooth manifold is called a ''smooth hypersurface''. In , a smooth hypersurface is orienta ... [...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] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thom Conjecture
In mathematics, a smooth algebraic curve C in the complex projective plane, of degree d, has Genus_(mathematics)#Topology, genus given by the genus–degree formula :g = (d-1)(d-2)/2. The Thom conjecture, named after French mathematician René Thom, states that if \Sigma is any smoothly embedded connected curve representing the same class in homology (mathematics), homology as C, then the genus g of \Sigma satisfies the inequality :g \geq (d-1)(d-2)/2. In particular, ''C'' is known as a ''genus minimizing representative'' of its homology class. It was first proved by Peter B. Kronheimer, Peter Kronheimer and Tomasz Mrowka in October 1994, using the then-new Seiberg–Witten invariants. Assuming that \Sigma has nonnegative self intersection number this was generalized to Kähler manifolds (an example being the complex projective plane) by John Morgan (mathematician), John Morgan, Zoltán Szabó (mathematician), Zoltán Szabó, and Clifford Taubes, also using the Seiberg–Wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enrico Arbarello
Enrico Arbarello is an Italian mathematician who is a leading expert in algebraic geometry. He earned a Ph.D. at Columbia University in New York in 1973. He was a visiting scholar at the Institute for Advanced Study from 1993-94. He is now a Mathematics Professor at Sapienza University of Rome. In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .... retrieved 2012-11-03. References External links [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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