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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.
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.
This is a list of useful examples in general topology, a field of mathematics. Alexandrov topology; Cantor space; Co-kappa topology Cocountable topology; Cofinite topology; Compact-open topology; Compactification; Discrete topology; Double-pointed cofinite topology; Extended real number line; Finite topological space; Hawaiian earring; Hilbert cube
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 ...
It is a finitely generated, torsion-free subgroup [20] and its study is of fundamental importance for its bearing on both the structure of the mapping class group itself (since the arithmetic group is comparatively very well understood, a lot of facts about boil down to a statement about its Torelli subgroup) and applications to 3 ...
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.
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.
For example, the genus of a smooth projective curve C which is an interesting invariant of the curve, is an integer, which can be read off the dimension of the first Betti cohomology group of C. So, the motive of the curve should contain the genus information.