Algebraic cycleIn mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety. The most trivial case is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one subvarieties, called divisors.
Conic sectionA conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions.
Singularity (mathematics)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 function has a singularity at , where the value of the function is not defined, as involving a division by zero. The absolute value function also has a singularity at , since it is not differentiable there. The algebraic curve defined by in the coordinate system has a singularity (called a cusp) at .
Ruled surfaceIn geometry, a surface S is ruled (also called a scroll) if through every point of S there is a straight line that lies on S. Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space. A ruled surface can be described as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle.
Euler sequenceIn mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an -fold sum of the dual of the Serre twisting sheaf. The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.) Let be the n-dimensional projective space over a commutative ring A. Let be the sheaf of 1-differentials on this space, and so on.
Adequate equivalence relationIn algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined intersection products. Pierre Samuel formalized the concept of an adequate equivalence relation in 1958. Since then it has become central to theory of motives. For every adequate equivalence relation, one may define the of pure motives with respect to that relation.
Commutative ringIn mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. A ring is a set equipped with two binary operations, i.e. operations combining any two elements of the ring to a third.
Classifying spaceIn mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e., a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG → BG. As explained later, this means that classifying spaces represent a set-valued functor on the of topological spaces.
Arithmetic of abelian varietiesIn mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety A over a number field K; or more generally (for global fields or more general finitely-generated rings or fields).
Uniformization theoremIn mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.
