Canonical curve

The image of an algebraic curve under a canonical imbedding. If a curve $X$ is not hyper-elliptic and has genus 2, then its image in the projective space $P^{g-1}$ under a canonical imbedding has degree $2g-2$ and is a normal curve. Conversely, any normal curve of degree $2g-2$ in $P^{g-1}$ is a canonical curve for some curve of genus $g$. Two algebraic curves (with the above condition) are birationally isomorphic if and only if their canonical curves are projectively equivalent. This reduces the problem of the classification of curves to that of the theory of projective invariants and provides the possibility of constructing a moduli variety of algebraic curves . For small $g$ it is possible to given an explicit geometric description of canonical curves of genus $g$. Thus, for genus 4 canonical curves are intersections of quadrics and cubics in $P^3$, while for genus 5 they are intersections of three quadrics in $P^4$.

Comments

The degree of a projective algebraic variety $V\subset P^g$ of dimension $n$ is the number of points of intersection with a generic hyperplane of dimension $g-n$ in $P^g$. Thus, the degree of a plane curve given by a homogeneous equation $f(X,Y,Z)=0$ in $P^2$ is equal to the degree of the polynomial $f$. See Algebraic curve for the definition of genus, and other notations occurring above.

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2a]	J.A. Dieudonné, "La géométrie des groupes classiques" , Springer (1955) Zbl 0221.20056
[2b]	D. Mumford, "Geometric invariant theory" , Springer (1965) MR0214602 Zbl 0147.39304
[3]	R.J. Walker, "Algebraic curves" , Springer (1978) MR0513824 Zbl 0399.14016
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[a1]	E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , 1 , Springer (1984) MR2807457 MR0770932 Zbl 05798333 Zbl 0991.14012 Zbl 0559.14017
[a2]	D. Mumford, "Curves and their Jacobians" , Univ. Michigan Press (1976) MR0419430 Zbl 0945.14001 Zbl 0316.14010