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The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k , where k is the non-orientable genus.
The genus-degree formula for plane curves can be deduced from the adjunction formula. [2] Let C ⊂ P 2 be a smooth plane curve of degree d and genus g. Let H be the class of a hyperplane in P 2, that is, the class of a line. The canonical class of P 2 is −3H.
Two different pants decompositions for the surface of genus 2. The importance of the pairs of pants in the study of surfaces stems from the following property: define the complexity of a connected compact surface of genus with boundary components to be () = +, and for a non-connected surface take the sum over all components.
The Jacobian of a curve over an arbitrary field was constructed by Weil (1948) as part of his proof of the Riemann hypothesis for curves over a finite field. The Abel–Jacobi theorem states that the torus thus built is a variety, the classical Jacobian of a curve, that indeed parametrizes the degree 0 line bundles, that is, it can be ...
Since these parameterizing functions are doubly periodic, the elliptic curve can be identified with a period parallelogram with the sides glued together i.e. a torus. So the genus of an elliptic curve is 1. The genus–degree formula is a generalization of this fact to higher genus curves. The basic idea would be to use higher degree equations.
More generally, the genus of a graph is the minimum genus of a two-dimensional surface into which the graph may be embedded; planar graphs have genus zero and nonplanar toroidal graphs have genus one. Every graph can be embedded without crossings into some (orientable, connected) closed two-dimensional surface (sphere with handles) and thus the ...
In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation.
In genus 2 it is a classical result that all such curves are hyperelliptic, [4] pg 298 so the moduli space can be determined completely from the branch locus of the curve using the Riemann–Hurwitz formula. Since an arbitrary genus 2 curve is given by a polynomial of the form () ()