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A stack may be implemented as, for example, a singly linked list with a pointer to the top element. A stack may be implemented to have a bounded capacity. If the stack is full and does not contain enough space to accept another element, the stack is in a state of stack overflow. A stack is needed to implement depth-first search.
For example, if a few points have non-trivial stabilisers, then the categorical quotient will not exist among schemes, 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 of schemes.
A stacky curve is a type of stack used in studying Gromov–Witten theory, enumerative geometry, and rings of modular forms. Stacky curves are closely related to 1-dimensional orbifolds and therefore sometimes called orbifold curves or orbicurves.
For example, if denotes the moduli stack of rank-n vector bundles, then there is a presentation given by the trivial bundle over (). A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism .
Another four-level stack interchange in the Baltimore area is located at the northeastern junction between I-695 and I-95. The stack was built as part of a massive I-95 reconstruction project that includes high-occupancy toll lanes (HOT lanes), designed to relieve congestion between Baltimore and its northeastern suburbs.
A differentiable stack is a stack : together with a special kind of representable submersion (every submersion described above is asked to be surjective), for some manifold . The map F X → C {\displaystyle F_{X}\to {\mathcal {C}}} is called atlas, presentation or cover of the stack X {\displaystyle X} .
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In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory.Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves, and the moduli stack of elliptic curves.