sol - PL ¨ UCKER FORMULAS Let f X → P n be a map of a...

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Unformatted text preview: PL ¨ UCKER FORMULAS Let f : X → P n be a map of a nonsingular projective curve X . It is given by an invertible sheaf L of some degree d and a basis ( s , . . . , s n ) of a linear subspace V of H ( X, L ). In coordinate-free way, we have a linear subspace V ⊂ H ( X, L ) of dimension n + 1 and the map X → P ( V * ) defined by f ( x )( s ) = P ( { s : s ( x ) = 0 } ) for any s ∈ V . For any point x ∈ X and s ∈ V \{ } we denote by ν x ( s ) the largest power of the maximal ideal m i X,x such that s x ∈ m i X,x L x . Let V i = ν- 1 x ( Z ≥ i ) , i ≥ 0. Since dim m i X,x / m i +1 X,x = 1, the quotient space V i /V i +1 is of dimension ≤ 1. This implies that there is a unique sequence of nonnegative numbers 0 ≤ α 1 ≤ . . . ≤ α n such that (1) { } ⊂ V n + α n ⊂ . . . ⊂ V 1+ α 1 ⊂ V, where codim V i + α i = i. The linear projective subspace P O i- 1 ( f, x ) = P ( V ⊥ i + α i ) of P ( V * ) of dimension i- 1 is called the osculating (i-1)-plane of f at the point x . The sequence of projective subspaces (2) P O ( f, x ) ⊂ . . . ⊂ P O n- 1 ( f, x ) ⊂ P ( V * ) is called the osculating flag of f at x with ramification sequence ( α 1 , . . . , α n ). Here the subscript indicates the dimension of the subspace. It is clear that P O ( f, x ) = f ( x ). Let us choose projective coordinates in such a way that (3) P O i ( f, x ) = { T i +1 = . . . = T n = 0 } , i = 0 , . . . , n- 1 . Let t i = T i /T be inhomogeneous coordinates, and let t be a local coordinate of X at x , the map f is given locally by (4) ( x 1 , . . . , x n ) = ( t 1+ α 1 g 1 ( t ) , t 2+ α 2 g 2 ( t ) , . . . , t n + α n g n ( t )) , where g i (0) 6 = 0. Under the homomorphism of ring k [ t 1 , . . . , t n ] → O P ( V * ) ,f ( x ) → O X,x the images of the functions t i belong to m i + α i X,x . In particular, if α 1 > 0, all functions belong to m 2 X,x and hence the map f is ramified at x . If α 1 = 0, and f- 1 ( f ( x )) = { x } , then f is a closed embedding in an affine neighborhood of x . In this case, the osculating hyperplane P O n- 1 ( f, x ) intersects f ( X ) at x with multiplicity n + α n . A point x is called a hyperosculating point if α n > 0. Clearly, all ramification points are hyperosculating points. An honest osculating point is a hyperosculating point with α 1 = . . . = α n- 1 = 0. Example 1. Assume n = 2 and let C ⊂ P 2 be an embedded plane curve with affine equation F ( x, y ) = 0 , F (0 , 0) = 0 . Let X be its normalization and f : X → C , → P 2 be the composition such that f ( x ) = (0 , 0) . Choose 1 2 PL ¨ UCKER FORMULAS the system coordinates as in (3) . Then the coordinate line x = 0 intersects the curve with multiplicity 1 + α 1 at the origin and the coordinate line y = 0 intersects the curve with multiplicity 2+ α 2 . An honest hyperosculating point with is called an inflection points (of order α n ). Obviously, C is singular at (0 , 0) if α 1 > 1 . If (0 , 0) is an ordinary cusp, then the osculating flag is...
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sol - PL ¨ UCKER FORMULAS Let f X → P n be a map of a...

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