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The genus–degree formula is a generalization of this fact to higher genus curves. The basic idea would be to use higher degree equations.
On a general Riemann surface of genus , has degree , independently of the meromorphic form chosen to represent the divisor. This follows from putting D = K {\displaystyle D=K} in the theorem. In particular, as long as D {\displaystyle D} has degree at least 2 g − 1 {\displaystyle 2g-1} , the correction term is 0, so that
However, some properties are not kept under birational equivalence and must be studied on non-plane curves. This is, in particular, the case for the degree and smoothness. For example, there exist smooth curves of genus 0 and degree greater than two, but any plane projection of such curves has singular points (see Genus–degree formula).
For genus g ≥ 3 it is no longer the case that the genus determines the gonality. The gonality of the generic curve of genus g is the floor function of (g + 3)/2. Trigonal curves are those with gonality 3, and this case gave rise to the name in general. Trigonal curves include the Picard curves, of genus three and given by an equation y 3 = Q(x)
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 first two invariants covered by the Plücker formulas are the degree d of the curve C and the degree d *, classically called the class of C. Geometrically, d is the number of times a given line intersects C with multiplicities properly counted. (This includes complex points and points at infinity since the curves are taken to be subsets of ...
Using the Riemann–Hurwitz formula, the hyperelliptic curve with genus g is defined by an equation with degree n = 2g + 2. Suppose f : X → P 1 is a branched covering with ramification degree 2, where X is a curve with genus g and P 1 is the Riemann sphere. Let g 1 = g and g 0 be the genus of P 1 ( = 0 ), then the Riemann-Hurwitz formula ...
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.