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52: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/06/27(土) 23:02:30.33 ID:jEjJjPRO

>>51
つづき

In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.

The most basic problem is that of moduli of smooth complete curves of a fixed genus. Over the field of complex numbers these correspond precisely to compact Riemann surfaces of the given genus, for which Bernhard Riemann proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure depends").

Genus 1
Main article: Moduli stack of elliptic curves

つづく

http://rio2016.5ch.net/test/read.cgi/math/1592654877/52

53: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/06/27(土) 23:03:08.86 ID:jEjJjPRO

>>52
つづき

Boundary geometry
Here the vertices of the graph correspond to irreducible components of the nodal curve, the labelling of a vertex is the arithmetic genus of the corresponding component, edges correspond to nodes of the curve and the half-edges correspond to the markings.
The closure of the locus of curves with a given dual graph in {\displaystyle {\overline {\mathcal {M}}}_{g,n}}\overline {{\mathcal  {M}}}_{{g,n}} is isomorphic to the stack quotient of a product {\displaystyle \prod _{v}{\overline {\mathcal {M}}}_{g_{v},n_{v}}}\prod _{v}\overline {{\mathcal  {M}}}_{{g_{v},n_{v}}} of compactified moduli spaces of curves by a finite group.
In the product the factor corresponding to a vertex v has genus gv taken from the labelling and number of markings {\displaystyle n_{v}}{\displaystyle n_{v}} equal to the number of outgoing edges and half-edges at v. The total genus g is the sum of the gv plus the number of closed cycles in the graph.

https://en.wikipedia.org/wiki/Moduli_stack_of_elliptic_curves
Moduli stack of elliptic curves
(引用終り)
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http://rio2016.5ch.net/test/read.cgi/math/1592654877/53

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