In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach...

vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may...

canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle...

In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common...

space plays a central role in algebraic geometry. This article aims to define the notion in terms of abstract algebraic geometry and to describe some...

\end{aligned}}} The curves γ x {\displaystyle \gamma _{x}} are called integral curves or trajectories (or less commonly, flow lines) of the vector field V {\displaystyle...

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as...

the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B. Many algebraic operations on real...

In algebraic geometry, a torsor or a principal bundle is an analogue of a principal bundle in algebraic topology. Because there are few open sets in Zariski...

concepts such as locally free modules, the algebraic counterpart to vector bundles. Roughly, affine spaces are vector spaces whose origins are not specified...

In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others...

This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory...

formal functions Ultrabornological space Weil conjectures Vector bundles on algebraic curves Zariski's main theorem Testimony by Pierre Cartier asserts...

information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under...

language of fiber bundles, such a map is called a section. A vector field on M {\displaystyle M} is therefore a section of the tangent bundle of M {\displaystyle...

function is required to be differentiable. Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one. A surface...

In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear...

formula). Weil's Riemann–Roch theorem for vector bundles on curves, and the Riemann–Roch theorem for algebraic surfaces (see below), are included in its...

tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of...

vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields...

line bundles and a generalization of the theorem to algebraic curves. The theorem will be illustrated by picking a point P {\displaystyle P} on the surface...

but it will exist as a stack. In the same way, moduli spaces of curves, vector bundles, or other geometric objects are often best defined as stacks instead...

specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves M g , n {\displaystyle...

Exterior algebras of vector bundles are frequently considered in geometry and topology. There are no essential differences between the algebraic properties...

In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent...

particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is...

smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then...

Deninger, Christopher; Werner, Annette (2007), "On Tannaka duality for vector bundles on p-adic curves", Algebraic cycles and motives. Vol. 2, London Math. Soc...

Whitney changed the name to sphere bundle. The theory of fibered spaces, of which vector bundles, principal bundles, topological fibrations and fibered...

In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line...