This page gathers the 66 dictionary entries belonging to the domain “Algebraic geometry”, as labelled by Wiktionary. Each word leads to its full entry: definitions, etymology, pronunciation, examples.
- adjoint noun A curve A such that any point of a given curve C of multiplicity r has multiplicity at least r–1 on A. Sometimes the multiple points of C…
- affine scheme noun An abstract, geometric object in mathematics which generalizes the notion of an affine variety qua domain of a commutative ring of…
- affine variety noun A set of points (in n-dimensional space) which satisfy a set of equations which have a polynomial of n variables on one side and a zero on…
- algebraic analysis noun The study of systems of linear partial differential equations, using techniques from sheaf theory and complex analysis to examine functions…
- algebraic function noun Informally, any function expressible using (only) the operations of addition, subtraction, multiplication, division and raising to a…
- algebraic variety noun The set of solutions of a given system of polynomial equations over the real or complex numbers; any of certain generalisations of such a…
- anabelian adj Of or relating to anabelian geometry, a proposed theory describing the way the algebraic fundamental group G of an algebraic variety (or…
- anabelian geometry noun A theory which describes the way in which the algebraic fundamental group G of an algebraic variety (or some related geometric object) V…
- B-sheaf noun A kind of partial, simpler to describe sheaf which can be extended into a sheaf (and which is, in a technical sense, equivalent to that…
- birational geometry noun A field of algebraic geometry in which the aim is to determine under what conditions two algebraic varieties are isomorphic outside…
- Calabi-Yau manifold noun A kind of manifold having properties, such as Ricci flatness, that yield applications in theoretical physics.
- Cartesian coordinates noun The coordinates of a point measured as its perpendicular distance to the right of a vertical axis (the y-axis) and its perpendicular…
- comb noun A connected and reduced curve with irreducible components consisting of a smooth subcurve (called the handle) and one or more additional…
- constructible adj Admitting description as a finite union of locally closed (with respect to the Zariski topology) sets.
- Cox ring noun A sort of universal homogeneous coordinate ring for a projective variety, and (roughly speaking) a direct sum of the spaces of sections of…
- cubic noun A cubic curve.
- curve noun An algebraic curve; a polynomial relation of the planar coordinates.
- distinguished open set noun A particularly basic kind of set, generalizing the notion of the complement of a hypersurface, originating in the study of algebraic…
- divisor noun An element of the free abelian group on the points of the space.
- Donaldson-Thomas invariant noun Given a compact moduli space of sheaves on a Calabi-Yau threefold, its Donaldson-Thomas invariant is the virtual number of its points…
- elliptic curve noun The algebraic set of an equation which equates the square of one variable to a reduced cubic polynomial in another variable, over some…
- flat adj Such that the induced map on every stalk is flat (as a map of rings).
- four-dimensional adj Having four dimensions; being measurable along four mutually perpendicular axes.
- Fourier-Mukai transform noun A functor Φ_K between derived category of coherent sheaves D(X) → D(Y) for schemes X and Y, which is, in a sense, an integral transform…
- fourth dimension noun The fourth coordinate indicating a position along the fourth axis.
- generic property noun A property that is true in some dense open subset of a given set.
- genus noun A natural number representing any of several related measures of the complexity of a given manifold or graph.
- geometry noun A mathematical object comprising representations of a space and of its spatial relationships.
- group functor noun A group object that is an object in a category of functors; a functor with certain properties that generalise the concept of group.
- handle noun The smooth, irreducible subcurve of a comb which connects to each of the other components in exactly one point.
- Iitaka dimension noun The Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective space…
- implicit function noun A function defined by a (multivariable) implicit equation when one of the variables is regarded as the value of the function, especially…
- intercept noun The coordinate of the point at which a curve intersects an axis.
- irreducible adj Inexpressible as the union of two proper algebraic subvarieties.
- isogeny noun An epimorphism of group schemes that is surjective and has a finite kernel.
- join noun The smallest linear space containing both of two given linear spaces.
- K-theory noun The study of rings R generated by the set of vector bundles over some topological space or scheme;
- local adj Such that the following conditions are equivalent: (1) P holds for R (M); (2) P holds for the localization R_p (M_p) for all prime ideals p…
- mirror symmetry noun A relationship between Calabi-Yau manifolds where two such manifolds look very different geometrically but are nevertheless equivalent when…
- nef adj Of a line bundle on a complete algebraic variety over a field: such that the degree of its restriction to every algebraic curve in the…
- noetherian adj admitting a finite covering by open affine subsets operatorname SpecA_i, where each A_i is a noetherian ring;
- nonic adj Describing a curve the equation of which is nonic.
- normal adj Such that the local ring at every point is an integrally closed domain.
- projective variety noun A Zariski closed subvariety of a projective space; the zero-locus of a set of homogeneous polynomials that generates a prime ideal.
- projectivization noun A process (more formally, a mapping) that, given a vector space, specifies an associated projective space; (loosely) the projective space…
- proper adj Separated, of finite type, and universally closed.
- quasi-projective adj Locally closed with respect to the Zariski topology on projective n-space; isomorphic to such a set.
- rational adj Of a variety: (informally) geometrically simple almost everywhere; (formally) birationally equivalent to projective space
- rational function noun Any function expressible as the quotient of two (coprime) polynomials (and which thus has poles at a finite, discrete set of points which…
- regular adj Such that the local ring at every point is regular.
- regular map noun A morphism between algebraic varieties.
- scheme theory noun The branch of mathematics that concerns schemes (locally ringed spaces admitting coverings by open sets, each isomorphic to the spectrum of…
- spectrum noun An abstract object in mathematics created from a commutative ring R and denoted operatorname Spec(R) or operatorname SpecR and said to be…
- squircle noun A closed quartic curve having properties intermediate between those of a square and a circle.
- tooth noun An irreducible component of a comb that intersects the handle in exactly one point, that point being distinct from the unique point of…
- toric adj Containing an algebraic torus as a dense subset, such that the group action of the torus on itself extends to the whole space; or, the…
- valence noun For a correspondence T on a curve: a number k such that the divisors T(P)+kP are all linearly equivalent.
- valency noun Alternative form of valence.
- vanishing ideal noun An ideal consisting of all polynomials (belonging to a certain polynomial ring) which zero out within a given algebraic variety.
- variety noun Ellipsis of algebraic variety (“the set of solutions of a given system of polynomial equations over the real or complex numbers; any of…
- Witt group noun given a variety X, the quotient of the Grothendieck group of isometry classes of quadratic spaces on X, with respect to orthogonal sum…
- y-axis noun The axis on a graph that is usually drawn from bottom to top and usually shows the range of values of variable dependent on one other…
- z-axis noun The axis on a graph of at least three dimensions that is usually drawn vertically and usually shows the range of values of a variable…
- Zariski topology noun Originally, a topology applicable to algebraic varieties, such that the closed sets are the variety's algebraic subvarieties; later, a…
- Zariski-Riemann space noun A locally ringed space of a subring k of a field K, whose points are valuation rings containing k and contained in K. They generalize the…
- zero locus noun The set of points (usually in affine or projective space, but more generally in the domain of the function) where the given polynomial (or…