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Moduli of algebraic curves
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<p>In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a moduli space of (algebraic) curves is a geometric space (typically a <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a> or an <a href="/facts/Algebraic_stack/flrtwvwU">algebraic stack</a>) whose points represent isomorphism classes of <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curves</a>. It is thus a special case of a <a href="/facts/Moduli_space/9sqrdpwR">moduli space</a>. 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 <a href="/facts/Moduli_space/9sqrdpwR">fine</a> and <a href="/facts/Moduli_space/9sqrdpwR">coarse moduli spaces</a> for the same moduli problem. </p><p>The most basic problem is that of moduli of <a href="/facts/Smooth_morphism/GNxTBHTG">smooth</a> <a href="/facts/Complete_variety/4rhVhqAU">complete</a> curves of a fixed <a href="/facts/Genus_(mathematics)/o1Yhf8ty">genus</a>. Over the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a> these correspond precisely to <a href="/facts/Compact_Riemann_surface/qoj8ogv8">compact Riemann surfaces</a> of the given genus, for which <a href="/facts/Bernhard_Riemann/oV2fL0k8">Bernhard Riemann</a> proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure depends"). </p>