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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 curve complex of a surface is a complex whose vertices are isotopy classes of simple closed curves on . The action of the mapping class groups Mod ( S ) {\displaystyle \operatorname {Mod} (S)} on the vertices carries over to the full complex.
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 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 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 ...
The most common examples are the curves X(N), X 0 (N), and X 1 (N) associated with the subgroups Γ(N), Γ 0 (N), and Γ 1 (N). The modular curve X(5) has genus 0: it is the Riemann sphere with 12 cusps located at the vertices of a regular icosahedron. The covering X(5) → X(1) is realized by the action of the icosahedral group on the Riemann ...