1.i. Ramification of Parameterized rational plane curves

    A parameterized rational plane curve   f : P¹ --> P²   of degree d is a composition of the rational normal curve   g : P¹ --> P^d   with a projection   p : P^d -->> P².   The centre of this projection is a (d-3)-plane H which does not meet the image of the rational normal curve g.   We say that the curve f is ramified at a point s of P¹ if the derivatives f(s), f'(s), f''(s) do not span P².   Thus, the (d-3)-plane H meets the 2-plane K₂(s) spanned by g(s), g'(s), and g''(s), and so Thus H satisfies the simple (codimension-1) Schubert condition imposed by K₂(s).   We expect H to satisfy 3(d-2) (= dimension of the Grassmannian of (d-3)-planes in P^d) such conditions, counted with multiplicity.

    This is in fact the case.   To see this, represent H as the column space of a (d+1) by (d-2) matrix, also denoted H.   Writing g and its derivatives as column vectors, K₂(s) is the concatenation of g(s), g'(s), and g''(s).   Then the inflection points of the parameterized rational plane curve   f :=f_H : P¹ -->> P²   are precisely the zeroes of the determinant

    Each inflection point s has a degree, which is multiplicity of that inflection; the order of vanishing of the determinant at the point s.   This invariant however is rather crude, and we introduce a finer invariant which measures just how linearly dependent are the derivatives of f.   The index of an inflection point s is a triple (a, b, c) defined as follows.

• a :=0.
• The derivatives f(s), ..., f^(b-1)(s) are proportional, but f^(b)(s) is not proportional to them.   Thus the first b+1 derivatives span a line l in P².
• The derivatives f(s), ..., f^(c-1)(s) all lie in l, but f^(c)(s) does not.

    We interpret this index geometrically.   Let K_j(s) be the j-plane spanned by the first (j+1) derivatives g(s), ... g^(j)(s) of the rational normal curve at the point s.   This is the j-plane osculating the rational normal curve at s.   Then a parameterized rational curve   f_H : P¹ --> P²   has index (a, b, c) when

• H contains K_a-1(s) but does not contain K_a(s).
• H intersects K_b-1(s) in a P^b-2 and intersects K_b(s) in that same P^b-2.
• H intersects K_c-1(s) in a P^c-3 and intersects K_c(s) in that same P^b-3.