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

constructing moduli spaces as algebraic stacks from moduli functors Moduli of algebraic curves Moduli stack of elliptic curves Moduli spaces of K-stable Fano...

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

specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities...

{\displaystyle T_{F}:=F(k[\varepsilon ]).} One of the first properties of the moduli of algebraic curves M g {\displaystyle {\mathcal {M}}_{g}} can be...

considered while constructing the Hilbert scheme of curves (and the moduli space of algebraic curves). This polynomial is H C ( t ) = ( 6 t − 1 ) ( g...

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...

algebraic stack over Spec ( Z ) {\displaystyle {\text{Spec}}(\mathbb {Z} )} classifying elliptic curves. Note that it is a special case of the moduli...

algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a...

new description of the moduli spaces for curves. Their work came to be seen as an introduction to one form of the theory of algebraic stacks, and recently...

Differential of the first kind Jacobian variety Generalized Jacobian Moduli of algebraic curves Hurwitz's theorem on automorphisms of a curve Clifford's...

{PGL}}(5g-6)]} Moduli of algebraic curves Stable map of curves Artin, M.; Winters, G. (1971-11-01). "Degenerate fibres and stable reduction of curves". Topology. 10...

abstract development of algebraic geometry. Over finite fields, étale cohomology provides topological invariants associated to algebraic varieties. p-adic...

(which, roughly, is an infinite product of classifying stacks.) Group-scheme action Moduli of algebraic curves The T-point is obtained by completing the...

Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC...

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...

variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity...

the number of moduli of a curve of genus g, unless g is 2. Much more is known about the hyperelliptic locus in the moduli space of curves or abelian varieties...

it was originally introduced to describe moduli of algebraic curves. A further generalization are the algebraic stacks, also called Artin stacks. DM stacks...

linear algebraic groups. stable 1.  A stable curve is a curve with some "mild" singularity, used to construct a good-behaving moduli space of curves. 2.  A...

complex curves at all. By contrast, an algebraic surface always contains many continuous families of curves.) Over an algebraically closed field of characteristic...

to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either...

modular group. The moduli space of algebraic curves M g {\displaystyle {\mathcal {M}}_{g}} defined as a universal family of smooth curves of given genus g...

Existence of the moduli scheme for curves of any genus. Mumford's work in geometry combined traditional geometric insights with the latest algebraic techniques...

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

referring to a singular curve. The modular function j(τ) is algebraic on imaginary quadratic numbers τ: these are the only algebraic numbers in the upper...

Comparison of computer algebra systems Isogeny j-line Level structure (algebraic geometry) Modularity theorem Moduli stack of elliptic curves Nagell–Lutz theorem...

study of moduli of curves, the important point is to understand the boundaries of the moduli, which amounts to understand degenerations of curves. Ruled-ness...

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

theory of divisors of tropical curves are related to chip-firing games on graphs associated to the tropical curves. Many classical theorems of algebraic geometry...