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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.
Consider a topological space, that is, a space with some notion of closeness between points in the space. We can consider the set of homeomorphisms from the space into itself, that is, continuous maps with continuous inverses: functions which stretch and deform the space continuously without breaking or gluing the space.
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.
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 ...
Colloquially speaking, the genus of a Riemann surface is its number of handles; for example the genus of the Riemann surface shown at the right is three. More precisely, the genus is defined as half of the first Betti number , i.e., half of the C {\displaystyle \mathbb {C} } -dimension of the first singular homology group H 1 ( X , C ...
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 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 P 2 {\displaystyle \mathbf {P} ^{2}} is a smooth curve of genus 2.)