In algebraic geometry, a moduli scheme is a moduli space that exists in the category of schemes developed by French mathematician Alexander Grothendieck...

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

generalized Kummer surface. Quot scheme Castelnuovo–Mumford regularity Matsusaka's big theorem Moduli of algebraic curves Moduli space Hilbert modular surface...

completed his PhD in 1961, with a thesis entitled Existence of the moduli scheme for curves of any genus. Mumford's work in geometry combined traditional...

In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism...

over any scheme Y. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as a moduli space. For...

scheme) X and provides techniques for forming the 'quotient' of X by G as a scheme with reasonable properties. One motivation was to construct moduli...

1007/BFb0091051. ISBN 978-3-540-10021-8. Chai, Ching-Li (1986). "Siegel Moduli Schemes and Their Compactifications over C {\displaystyle \mathbb {C} } ". Arithmetic...

Concept in algebraic geometry Nakai conjecture Moduli scheme – Moduli space in the Grothendieck category of schemes Motive (algebraic geometry) – Structure for...

points collided. The Hilbert scheme Hilb ⁡ ( k , d , n ) {\displaystyle \operatorname {Hilb} (k,d,n)} is the fine moduli scheme of closed subschemes of dimension...

_{X}:X\to \mathbb {Z} } such that if X is a quasi-projective proper moduli scheme carrying a symmetric obstruction theory, then the weighted Euler characteristic...

elliptic curves, the j-line over a ring R is the coarse moduli scheme attached to the moduli problem sending a ring R {\displaystyle R} to the set of...

\mathbb {P} _{S}^{5g-6}} . Using the standard Hilbert scheme theory we can construct a moduli scheme of curves of genus g {\displaystyle g} embedded in some...

{\displaystyle S} . For any scheme (or S {\displaystyle S} -scheme) X {\displaystyle X} , the X {\displaystyle X} -points of the moduli stack are the groupoid...

algebraic geometry; for now see also ind-scheme). moduli See for example moduli space. While much of the early work on moduli, especially since [Mum65], put the...

Noetherian, such as the Moduli of algebraic curves and Moduli of stable vector bundles. Also, this property can be used to show many schemes considered in algebraic...

main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations...

contexts of Galois representations and moduli problems. The initial development of the theory of group schemes was due to Alexander Grothendieck, Michel...

playing the role of a moduli stack for higher-dimensional abelian varieties. One can solve this problem by constructing a moduli stack of abelian varieties...

{\displaystyle C,\pi :C\to \mathbf {P} ^{1}} ) where C is a smooth curve of genus g and π has degree d. Joe Harris and Ian Morrison. Moduli of curves. v t e...

generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques...

A^{\vee }} (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrised by a k-variety T...

Hilbert scheme is a scheme rather than a stack, because, very roughly speaking, deformation theory is simpler for closed schemes.) Some moduli problems...

parametrized by a moduli space. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem...

symmetry reduction scheme. Other such master theories are four-dimensional Chern–Simons theory and the affine Gaudin model. The moduli space of Yang–Mills...

family is either a Hilbert scheme or Quot scheme, or a quotient of one of them. For example, in the construction of the moduli of curves, it is constructed...

In mathematics, formal moduli are an aspect of the theory of moduli spaces (of algebraic varieties or vector bundles, for example), closely linked to deformation...

conditions are used in the construction of the moduli stack of elliptic curves and the construction of the moduli stack of pointed curves. Throughout this article...

group; i.e., the moduli stack was the moduli stack of vector bundles, but, today, the term refers to any of generalizations. The scheme-theoretic version...

In algebraic geometry, the moduli stack of rank-n vector bundles Vectn is the stack parametrizing vector bundles (or locally free sheaves) of rank n over...