# Canonical curve

The image of an algebraic curve under a canonical imbedding. If a curve is not hyper-elliptic and has genus 2, then its image in the projective space under a canonical imbedding has degree and is a normal curve. Conversely, any normal curve of degree in is a canonical curve for some curve of genus . 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 it is possible to given an explicit geometric description of canonical curves of genus . Thus, for genus 4 canonical curves are intersections of quadrics and cubics in , while for genus 5 they are intersections of three quadrics in .

#### References

 [1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 [2a] J.A. Dieudonné, "La géométrie des groups 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 [4] F. Severi, "Vorlesungen über algebraische Geometrie" , Teubner (1921) MR0245574 Zbl 48.0687.01