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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.
This is because the surface N has a unique class of one-sided curves such that, when N is cut open along such a curve C, the resulting surface is a torus with a disk removed. As an unoriented surface, its mapping class group is GL ( 2 , Z ) {\displaystyle \operatorname {GL} (2,\mathbb {Z} )} .
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.
The curve has genus one (genus formula); in particular, it is not isomorphic to the projective line P 1, which has genus zero. Using genus to distinguish curves is very basic: in fact, the genus is the first invariant one uses to classify curves (see also the construction of moduli of algebraic curves).
It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves. The mapping class group can be defined for arbitrary manifolds (indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in group theory .
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 ...
The double cover X of the projective plane branched along a smooth sextic (degree 6) curve is a K3 surface of genus 2 (that is, degree 2g−2 = 2). (This terminology means that the inverse image in X of a general hyperplane in is a smooth curve of genus 2.)
The algebraic definition of genus agrees with the topological notion. On a nonsingular curve, the canonical line bundle has degree 2g − 2. The notion of genus features prominently in the statement of the Riemann–Roch theorem (see also Riemann–Roch theorem for algebraic curves) and of the Riemann–Hurwitz formula.