# Algebraic geometry

## Algebraic geometry Topics

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### Projectivization

Given a vector space , its projectivization , sometimes written , is the set of equivalence classes for any in . For example, complex projective space has homogeneous coordinates , with not all .The projectivization is a manifold with one less dimension than . In fact, it is covered by the affine coordinate charts,

### K&auml;hler potential

The Kähler potential is a real-valued function on a Kähler manifold for which the Kähler form can be written as . Here, the operators(1)and(2)are called the del and del bar operator, respectively.For example, in , the function is a Kähler potential for the standard Kähler form, because(3)(4)(5)(6)

### Twistor correspondence

Oriented spheres in complex Euclidean three-space can be represented as lines in complex projective three-space ("Lie correspondence"), and the spheres may be thought of as the representation of the light cones of events in Minkowski space. In effect, the Lie correspondence represents the points of (complexified compactified) Minkowski space by lines in complex projective three-space, where meeting lines describe null-separated Minkowski points. This is the twistor correspondence.

### K&auml;hler identities

A collection of identities which hold on a Kähler manifold, also called the Hodge identities. Let be a Kähler form, be the exterior derivative, where is the del bar operator, be the commutator of two differential operators, and denote the formal adjoint of . The following operators also act on differential forms on a Kähler manifold:(1)(2)(3)where is the almost complex structure, , and denotes the interior product. Then(4)(5)(6)(7)(8)(9)In addition,(10)(11)(12)(13)These identities have many implications. For instance, the two operators(14)and(15)(called Laplacians because they are elliptic operators) satisfy . At this point, assume that is also a compact manifold. Along with Hodge's theorem, this equality of Laplacians proves the Hodge decomposition. The operators and commute with these Laplacians. By Hodge's theorem, they act on cohomology, which is represented by harmonic forms. Moreover, defining(16)where..

### Pl&uuml;cker's equations

Relationships between the number of singularitiesof plane algebraic curves. Given a plane curve,(1)(2)(3)(4)where is the class, the curve order, the number of ordinary double points, the number of cusps, the number of inflection points (inflection points), and the number of bitangents. Only three of these equations are linearly independent.

### Pl&uuml;cker characteristics

The class , curve order , number of ordinary double points , number of cusps , number of inflection points (inflection points) , number of bitangents , and curve genus .

### Zeuthen's theorem

If there is a correspondence between two curves of curve genus and and the number of branch points properly counted are and , then

### Piriform surface

A generalization to a quartic three-dimensional surface is the quartic surface of revolution(1)illustrated above. With , this surface is termed the "zeck" surface by Hauser. It has volume(2)geometric centroid(3)(4)(5)and inertia tensor(6)for constant density and mass .

### Piriform curve

A quartic algebraic curve also called the peg-top curve and given by the Cartesian equation(1)and the parametric curves(2)(3)for . It was studied by G. de Longchamps in 1886.The area of the piriform is(4)which is exactly the same as the ellipse with semiaxes and .The curvature of the piriform is given by(5)

### Weierstrass's gap theorem

Given a succession of nonsingular points which are on a nonhyperelliptic curve of curve genus , but are not a group of the canonical series, the number of groups of the first which cannot constitute the group of simple poles of a rational function is . If points next to each other are taken, then the theorem becomes: Given a nonsingular point of a nonhyperelliptic curve of curve genus , then the orders which it cannot possess as the single pole of a rational function are in number.

### Ordinary double point

An ordinary double point of a plane curve is point where a curve intersects itself such that two branches of the curve have distinct tangent lines. Ordinary double points of plane curves are commonly known as crunodes. Ordinary double points of a plane curves given by satisfy(1)where denotes a partial derivative.Let (or ) be a space curve. Then a point (where denotes the immersion of ) is an ordinary double point of the space curve if its preimage under consists of two values and , and the two tangent vectors and are noncollinear. Geometrically, this means that, in a neighborhood of , the curve consists of two transverse branches. Ordinary double points are isolated singularities having Coxeter-Dynkin diagram of type , and also called "nodes" or "simple double points."Ordinary double points of a surface given by satisfy(2)where denotes a partial derivative. A surface in complex three-space admits at most finitely many..

### Curve genus

One of the Plücker characteristics,defined bywhere is the class, the order, the number of nodes, the number of cusps, the number of stationary tangents (inflection points), and the number of bitangents.

### Noether's fundamental theorem

If two curves and of multiplicities and have only ordinary points or ordinary singular points and cusps in common, then every curve which has at least multiplicityat every point (distinct or infinitely near) can be writtenwhere the curves and have multiplicities at least and .

### Chasles's theorem

If two projective pencils of curves of orders and have no common curve, the locus of the intersections of corresponding curves of the two is a curve of order through all the centers of either pencil. Conversely, if a curve of order contains all centers of a pencil of order to the multiplicity demanded by Noether's fundamental theorem, then it is the locus of the intersections of corresponding curves of this pencil and one of order projective therewith.

### Study's theorem

Given three curves , , with the common group of ordinary points (which may be empty), let their remaining groups of intersections , , and also be ordinary points. If is any other curve through , then there exist two other curves , such that the three combined curves are of the same order and linearly dependent, each curve contains the corresponding group , and every intersection of or with or lies on or .

### Spinode

A function has a spinode (also called a horizontal cusp) at a point if is continuous at andfrom one side whilefrom the other side, so the curve is continuousbut the derivative is not.s

### Klein quartic

Consider the plane quartic curve defined bywhere homogeneous coordinates have been used here so that can be considered a parameter (the plot above shows the curve for a number of values of between and 2), over a field of characteristic 3. Hartshorne (1977, p. 305) terms this "a funny curve" since it is nonsingular, every point is an inflection point, and the dual curve is isomorphic to but the natural map is purely inseparable.The surface in complex projective coordinates (Levy 1999, p. ix; left figure), and with the ideal surface determined by the equation(Thurston 1999, p. 3; right figure) is more properly known as the Klein quarticor Klein curve. It has constant zero Gaussian curvature.Klein (1879; translation reprinted in 1999) discovered that this surface has a number of remarkable properties, including an incredible 336-fold symmetry when mirror reflections are allowed (Levy 1999, p. ix; Thurston..

### Cayleyian curve

The envelope of the lines connecting corresponding points on the Jacobian curve and Steinerian curve. The Cayleyian curve of a net of curves of order has the same curve genus as the Jacobian curve and Steinerian curve and, in general, the class .

### Klein's equation

If a real algebraic curve has no singularities except nodes and cusps, bitangents, and inflection points, thenwhere is the order, is the number of conjugate tangents, is the number of real inflections, is the class, is the number of real conjugate points, and is the number of real cusps. This is also called Klein's theorem.

### Rosatti's theorem

There is a one-to-one correspondence between the sets of equivalent correspondences (not of value 0) on an irreducible curve of curve genus , and the rational collineations of a projective space of dimensions which leave invariant a space of dimensions. The number of linearly independent correspondences will be that of linearly independent collineations.

### Bifolium

The bifolium is a folium with . The bifolium is a quartic curve and is given by the implicit equation is(1)and the polar equation(2)The bifolium has area(3)(4)(5)Its arc length is(6)(7)(OEIS A118307), where , , , and are elliptic integrals with(8)(9)The curvature is given by(10)(11)The bifolium is the pedal curve of the deltoid where the pedal point is the midpoint of one of the three curved sides.

### B&eacute;zout's theorem

Bézout's theorem for curves states that, in general, two algebraic curves of degrees and intersect in points and cannot meet in more than points unless they have a component in common (i.e., the equations defining them have a common factor; Coolidge 1959, p. 10).Bézout's theorem for polynomials states that if and are two polynomials with no roots in common, then there exist two other polynomials and such that . Similarly, given polynomial equations of degrees , , ... in variables, there are in general common solutions.Séroul (2000, p. 10) uses the term Bézout's theorem for the following two theorems. 1. Let be any two integers, then there exist such that2. Two integers and are relatively prime if there exist such that

### Riemann's integral theorem

Associated with an irreducible curve of curve genus , there are linearly independent integrals of the first sort. The roots of the integrands are groups of the canonical series, and every such group will give rise to exactly one integral of the first sort.

### Isolated point

An isolated point of a graph is a node of degree 0 (Hartsfield and Ringel 1990, p. 8; Harary 1994, p. 15; D'Angelo and West 2000, p. 212; West 2000, p. 22). The number of -node graphs with no isolated points are 0, 1, 2, 7, 23, 122, 888, ... (OEIS A002494), the first few of which are illustrated above. The number of graphical partitions of length is equal to the number of -node graphs that have no isolated points.Connected graphs have no isolated points.An isolated point on a curve is more commonly known as an acnode.An isolated point of a discrete set is a member of (Krantz 1999, p. 63).

### Humbert's theorem

The necessary and sufficient condition that an algebraic curve has an algebraic involute is that the arc length is a two-valued algebraic function of the coordinates of the extremities. Furthermore, this function is a root of a quadratic equation whose coefficients are rational functions of and .

### Rational point

A -rational point is a point on an algebraic curve , where and are in a field . For example, rational point in the field of ordinary rational numbers is a point satisfying the given equation such that both and are rational numbers.The rational point may also be a point at infinity.For example, take the elliptic curve(1)and homogenize it by introducing a third variable so that each term has degree 3 as follows:(2)Now, find the points at infinity by setting , obtaining(3)Solving gives , equal to any value, and (by definition) . Despite freedom in the choice of , there is only a single point at infinity because the two triples (, , ), (, , ) are considered to be equivalent (or identified) only if one is a scalar multiple of the other. Here, (0, 0, 0) is not considered to be a valid point. The triples (, , 1) correspond to the ordinary points (, ), and the triples (, , 0) correspond to the points at infinity, usually called the line at infinity.The rational points on elliptic..

### Hilbert polynomial

Let be an algebraic curve in a projective space of dimension , and let be the prime ideal defining , and let be the number of linearly independent forms of degree modulo . For large , is a polynomial known as the Hilbert polynomial.

A curve which has at least multiplicity at each point where a given curve (having only ordinary singular points and cusps) has a multiplicity is called the adjoint to the given curve. When the adjoint curve is of order , it is called a special adjoint curve.

### Acnode

An acnode, also called an isolated point or hermit point, of a curve is a point that satisfies the equation of the curve but has no neighboring points that also lie on the curve. The plot above shows the curve , which has an acnode at the origin .The conchoid of de Sluze has an acnode at the origin for .

### Abel's curve theorem

The sum of the values of an integral of the "first"or "second" sortandfrom a fixed point to the points of intersection with a curve depending rationally upon any number of parameters is a rational function of those parameters.

### Affine variety

An affine variety is an algebraic variety contained in affine space. For example,(1)is the cone, and(2)is a conic section, which is a subvariety of the cone. The cone can be written to indicate that it is the variety corresponding to . Naturally, many other polynomials vanish on , in fact all polynomials in . The set is an ideal in the polynomial ring . Note also, that the ideal of polynomials vanishing on the conic section is the ideal generated by and .A morphism between two affine varieties is given by polynomial coordinate functions. For example, the map is a morphism from to . Two affine varieties are isomorphic if there is a morphism which has an inverse morphism. For example, the affine variety is isomorphic to the cone via the coordinate change .Many polynomials may be factored, for instance , and then . Consequently, only irreducible polynomials, and more generally only prime ideals are used in the definition of a variety. An affine variety is..

### K&auml;hler structure

A Kähler structure on a complex manifold combines a Riemannian metric on the underlying real manifold with the complex structure. Such a structure brings together geometry and complex analysis, and the main examples come from algebraic geometry. When has complex dimensions, then it has real dimensions. A Kähler structure is related to the unitary group , which embeds in as the orthogonal matrices that preserve the almost complex structure (multiplication by ''). In a coordinate chart, the complex structure of defines a multiplication by and the metric defines orthogonality for tangent vectors. On a Kähler manifold, these two notions (and their derivatives) are related.The following are elements of a Kähler structure, with each condition sufficientfor a Kähler structure to exist.1. A Kähler metric. Near any point , there exists holomorphic coordinates such that the metric has the form(1)where denotes..

### K&auml;hler manifold

A complex manifold for which the exterior derivative of the fundamental form associated with the given Hermitian metric vanishes, so . In other words, it is a complex manifold with a Kähler structure. It has a Kähler form, so it is also a symplectic manifold. It has a Kähler metric, so it is also a Riemannian manifold.The simplest example of a Kähler manifold is a Riemann surface, which is a complex manifold of dimension 1. In this case, the imaginary part of any Hermitian metric must be a closed form since all 2-forms are closed on a two real dimensional manifold.

### Quaternion k&auml;hler manifold

A quaternion Kähler manifold is a Riemannian manifold of dimension , , whose holonomy is, up to conjugacy, a subgroup ofbut is not a subgroup of . These manifolds are sometimes called quaternionic Kähler and are sometimes written hyphenated as quaternion-Kähler, quaternionic-Kähler, etc.Despite their name, quaternion-Kähler manifolds need not be Kähler due to the fact that all Kähler manifolds have holonomy groups which are subgroups of , whereas . Depending on the literature, such manifolds are sometimes assumed to be connected and/or orientable. In the above definition, the case for is usually excluded due to the fact that which, under Berger's classification of holonomy, implies merely that the manifold is Riemannian. The above classification can be extended to the case where by requiring that the manifold be both an Einstein manifold and self-dual.Some authors exclude this last criterion,..

### K&auml;hler metric

A Kähler metric is a Riemannian metric on a complex manifold which gives a Kähler structure, i.e., it is a Kähler manifold with a Kähler form. However, the term "Kähler metric" can also refer to the corresponding Hermitian metric , where is the Kähler form, defined by . Here, the operator is the almost complex structure, a linear map on tangent vectors satisfying , induced by multiplication by . In coordinates , the operator satisfies and .The operator depends on the complex structure, and on a Kähler manifold, it must preserve the Kähler metric. For a metric to be Kähler, one additional condition must also be satisfied, namely that it can be expressed in terms of the metric and the complex structure. Near any point , there exists holomorphic coordinates such that the metric has the formwhere denotes the vector space tensor product; that is, it vanishes up to order two at . Hence, any geometric..

### Hilbert function

Given a finitely generated -graded module over a graded ring (finitely generated over , which is an Artinian local ring), the Hilbert function of is the map such that, for all ,(1)where denotes the length. If is the dimension of , then there exists a polynomial of degree with rational coefficients (called the Hilbert polynomial of ) such that for all sufficiently large .The power series(2)is called the Hilbert series of . It is a rational function that can be written in a unique way in the form(3)where is a finite linear combination with integer coefficients of powers of and . If is positively graded, i.e., for all , then is an ordinary polynomial with integer coefficients in the variable . If moreover , then , i.e., the Hilbert series is a polynomial.If has a finite graded free resolution(4)then(5)Moreover, if is a regular sequence over of homogeneous elements of degree 1, then the Hilbert function of the -dimensional quotient module is(6)and in particular,(7)These..

### Gerbe

There are no fewer than two closely related but somewhat different notions of gerbe in mathematics.For a fixed topological space , a gerbe on can refer to a stack of groupoids on satisfying the properties 1. for subsets open, and 2. given objects , any point has a neighborhood for which there is at least one morphism in . The second definition is due to Giraud (Brylinski 1993). Given a manifold and a Lie group , a gerbe with band is a sheaf of groupoids over satisfying the following three properties: 1. Given any object of , the sheaf of automorphisms of this object is a sheaf of groups on which is locally isomorphic to the sheaf of smooth -valued functions. Such a local isomorphism is unique up to inner automorphisms of . 2. Given two objects and of , there exists a surjective local homeomorphism such that and are isomorphic. In particular, and are locally isomorphic. 3. There exists a surjective local homeomorphism such that the category is non-empty. Clearly,..

### Band

A band over a fixed topological space is represented by a cover , , and for each , a sheaf of groups on along with outer automorphisms satisfying the cocycle conditions and . Here, restrictions of the cover to a finer cover should be viewed as defining the exact same band.The collection of all bands over the space with respect to a single cover has a natural category structure. Indeed, if and are two bands over with respect to , then an isomorphism consists of outer automorphisms compatible on overlaps so that . The collection of all such bands and isomorphisms thereof forms a category.The notion of band is essential to the study of gerbes (Moerdijk). In particular, for a gerbe over a topological space , one can choose an open cover of by open subsets , and for each , an object which together form a sheaf of groups on . One can then consider a collection of sheaf isomorphisms between any two groups and which forms a collection of well-defined outer automorphisms.In..

### Affine coordinates

The coordinates representing any point of an -dimensional affine space by an -tuple of real numbers, thus establishing a one-to-one correspondence between and .If is the underlying vector space, and is the origin, every point of is identified with the -tuple of the components of vector with respect to a given basis of .If is a three-dimensional space, each basis can be depicted by choosing its elements as the unit vectors of the -axis, the -axis, and the -axis, respectively. In general, this will produce three axes which are not necessarily perpendicular, and where the units are set differently. Hence, Cartesian coordinates are a very special kind of affine coordinates that correspond to the case where , , .

### K&auml;hler form

A closed two-form on a complex manifold which is also the negative imaginary part of a Hermitian metric is called a Kähler form. In this case, is called a Kähler manifold and , the real part of the Hermitian metric, is called a Kähler metric. The Kähler form combines the metric and the complex structure, indeed(1)where is the almost complex structure induced by multiplication by . Since the Kähler form comes from a Hermitian metric, it is preserved by , i.e., since . The equation implies that the metric and the complex structure are related. It gives a Kähler structure, and has many implications.On , the Kähler form can be written as(2)(3)where . In general, the Kähler form can be written in coordinates(4)where is a Hermitian metric, the real part of which is the Kähler metric. Locally, a Kähler form can be written as , where is a function called a Kähler potential. The Kähler form is..

### Birkhoff's theorem

Let and be two algebras over the same signature , with carriers and , respectively (cf. universal algebra). is a subalgebra of if and every function of is the restriction of the respective function of on .The (direct) product of algebras and is an algebra whose carrier is the Cartesian product of and and such that for every and all and all ,A nonempty class of algebras over the same signature is called a variety if it is closed under subalgebras, homomorphic images, and Cartesian products over arbitrary families of structures belonging to the class.A class of algebras is said to satisfy the identity if this identity holds in every algebra from this class. Let be a set of identities over signature . A class of algebras over is called an equational class if it is the class of algebras satisfying all identities from . In this case, is said to be axiomatized by .Birkhoff's theorem states that is an equational class iff it is a variety...

### Toric variety

Let , , ..., be distinct primitive elements of a two-dimensional lattice such that for , ..., . Each collection then forms a set of rays of a unique complete fan in , and therefore determines a two-dimensional toric variety .

### Algebraic variety

An algebraic variety is a generalization to dimensions of algebraic curves. More technically, an algebraic variety is a reduced scheme of finite type over a field . An algebraic variety in (or ) is defined as the set of points satisfying a system of polynomial equations for , 2, .... According to the Hilbert basis theorem, a finite number of equations suffices.A variety is the set of common zeros to a collection of polynomials. In classical algebraic geometry, the polynomials have complex numbers for coefficients. Because of the fundamental theorem of algebra, such polynomials always have zeros. For example,is the cone, andis a conic section, which is a subvarietyof the cone.Actually, the cone and the conic section are examples of affine varieties because they are in affine space. A general variety is comprised of affine varieties glued together, like the coordinate charts of a manifold. The field of coefficients can be any algebraically closed..

### Algebraic set

An algebraic set is the locus of zeros of a collection of polynomials. For example, the circle is the set of zeros of and the point at is the set of zeros of and . The algebraic set is the set of solutions to . It decomposes into two irreducible algebraic sets, called algebraic varieties. In general, an algebraic set can be written uniquely as the finite union of algebraic varieties.The intersection of two algebraic sets is an algebraic set corresponding to the union of the polynomials. For example, and intersect at , i.e., where and . In fact, the intersection of an arbitrary number of algebraic sets is itself an algebraic set. However, only a finite union of algebraic sets is algebraic. If is the set of solutions to and is the set of solutions to , then is the set of solutions to . Consequently, the algebraic sets are the closed sets in a topology, called the Zariski topology.The set of polynomials vanishing on an algebraic set is an ideal in the polynomial ring...

### Subvariety

A subset of an algebraic variety which is itself a variety. Every variety is a subvariety of itself; other subvarieties are called proper subvarieties.A sphere of the three-dimensional Euclidean space is an algebraic variety since it is defined by a polynomial equation. For example,(1)defines the sphere of radius 1 centered at the origin. Its intersection with the -plane is a circle given by the system of polynomial equations:(2)(3)Hence the circle is itself an algebraicvariety, and a subvariety of the sphere, and of the plane as well.Whenever some new independent equations are added to the equations defining a certain variety, the resulting variety will be smaller, since its points will be subject to more conditions than before. In the language of ring theory, this means that, while the sphere is the zero set of all polynomials of the ideal of , every subvariety of it will be defined by a larger ideal; this is the case for , which is the defining..

### Schubert variety

A class of subvarieties of the Grassmannian . Given integers , the Schubert variety is the set of points of representing the -dimensional subspaces of such that, for all ,It is a projective algebraic varietyof dimension

### Coordinate ring

Given an affine variety in the -dimensional affine space , where is an algebraically closed field, the coordinate ring of is the quotient ringwhere is the ideal formed by all polynomials with coefficients in which are zero at all points of . If is the entire -dimensional affine space , then this ideal is the zero ideal. It follows that the coordinate ring of is the polynomial ring . The coordinate ring of a plane curve defined by the Cartesian equation in the affine plane is .In general, the Krull dimension of ring is equal to the dimension of as a closed set of the Zariski topology of .Two polynomials and define the same function on iff . Hence the elements of are equivalence classes which can be identified with the polynomial functions from to .

### Affine scheme

Let be the set of prime ideals of a commutative ring . Then an affine scheme is a technical mathematical object defined as the ring spectrum of , regarded as a local-ringed space with a structure sheaf. A local-ringed space that is locally isomorphic to an affine scheme is called a scheme (Itô 1986, p. 69). An affine scheme is a generalization of the notion of affine variety, where the coordinate ring is replaced by any commutative unit ring, and the variety with the Zariski topology is replaced by any topological space.

### Picard variety

Let be a variety, and write for the set of divisors, for the set of divisors linearly equivalent to 0, and for the group of divisors algebraically equal to 0. Then is called the Picard variety. The Albanese variety is dual to the Picard variety.

### Chow coordinates

A generalization of Grassmann coordinates to -D algebraic varieties of degree in , where is an -dimensional projective space. To define the Chow coordinates, take the intersection of an -D algebraic variety of degree by an -D subspace of . Then the coordinates of the points of intersection are algebraic functions of the Grassmann coordinates of , and by taking a symmetric function of the algebraic functions, a homogeneous polynomial known as the Chow form of is obtained. The Chow coordinates are then the coefficients of the Chow form. Chow coordinates can generate the smallest field of definition of a divisor.

### Algebraic geometry

Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and the geometry is the set of zeros of polynomials, called an algebraic variety. For instance, the unit circle is the set of zeros of and is an algebraic variety, as are all of the conic sections.In the twentieth century, it was discovered that the basic ideas of classical algebraic geometry can be applied to any commutative ring with a unit, such as the integers. The geometry of such a ring is determined by its algebraic structure, in particular its prime ideals. Grothendieck defined schemes as the basic geometric objects, which have the same relationship to the geometry of a ring as a manifold to a coordinate chart. The language of category theory evolved at around the same time, largely in response to the needs of the increasing abstraction in algebraic geometry.As a consequence,..

### Algebraic curve

An algebraic curve over a field is an equation , where is a polynomial in and with coefficients in . A nonsingular algebraic curve is an algebraic curve over which has no singular points over . A point on an algebraic curve is simply a solution of the equation of the curve. A -rational point is a point on the curve, where and are in the field .The following table lists the names of algebraic curves of a given degree.ordercurveexamples2quadratic curvecircle, ellipse, hyperbola, parabola3cubic curvecissoid of Diocles, conchoid of de Sluze, folium of Descartes, Maclaurin trisectrix, Maltese cross curve, Mordell curve, Ochoa curve, right strophoid, semicubical parabola, serpentine curve, Tschirnhausen cubic, witch of Agnesi4quartic curveampersand curve, bean curve, bicorn, bicuspid curve, bifoliate, bifolium, bitangent-rich curve, bow, bullet nose, butterfly curve, capricornoid, cardioid, Cartesian ovals, Cassini ovals, conchoid..