Moduli of algebraic curves

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"). Moduli stacks of stable curves The moduli stack \mathcal{M}_{g} classifies families of smooth projective curves, together with their

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We describe an injection from border-strip decompositions of certain diagrams to permutations. This allows us to provide enumeration results as well as q-analogues of enumeration formulas. Finally, we use this injection to prove a connection between the number of border-strip decompositions of the n x 2n rectangle and the Weil-Petersson volume of the moduli space of an n-punctured Riemann sphere.

2019

Ate Pairing on Hyperelliptic Curves

Robert Granger

In this paper we show that the Ate pairing, originally defined for elliptic curves, generalises to hyperelliptic curves and in fact to arbitrary algebraic curves. It has the following surprising properties: The loop length in Miller’s algorithm can be up to g times shorter than for the Tate pairing, with g the genus of the curve, and the pairing is automatically reduced, i.e. no final exponentiation is needed.

Springer-Verlag Berlin Heidelberg2007

MATH-410: Riemann surfaces

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic curves.

Scheme (mathematics)

In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the

Alexander Grothendieck

Alexander Grothendieck (ˈgroʊtəndiːk; ˌalɛˈksandɐ ˈɡʁoːtn̩ˌdiːk; ɡʁɔtɛndik; 28 March 1928 – 13 November 2014) was a French mathematician who became the leading figure in the creation of modern al

Moduli space

In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or