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56 2. Plane Algebraic Curves

Exercises

1. For k infinite prove that P is a point of order ≤ r on a hypersurface H f of degree n if
and only if there exists a line L through P intersecting H f at n − r additional points.
2. In the [m(m + 3)/2]-dimensional family H m of all curves of degree m, calculate the
2
dimension of the subfamily of all curves passing through a point P and having order ≥ r
m
at that point. Give a lower bound for the dimension of the subfamily of all curves in H
2
passing through the points P i and having order ≥ r i at P i for i = 1,... , t.
3. Let C f and C g be two plane algebraic curves of degrees m and n, respectively, where
m n
f (w, x, y) = a 0 y + ··· + a m , g(w, x, y) = b 0 y + ··· + b n with a i = a i (w, x)
and b j = b j (w, x) homogeneous polynomials of degree i. j.If w : x : y is a point on
C f ∩ C g with order r on C f and order s on C g , then show that R( f, g)(W, X) has a
sr
factor of the form (xW − wX) in its factorization as a product of linear forms.
4. Let C f and C g be two plane algebraic curves of degrees m and n, respectively, without
common factors. If P 1 ,... , P t are the intersection points where r i is the order of C f at
P i and s i is the order of C g at P i , then show that the following relation holds

r i s i ≤ mn.
1≤i≤t
5. Let C f be a curve of degree m with no multiple components and with orders r i of the
singular points P i . Then show

m(m − 1) ≥ r i (r i − 1).
Also, show that this is the best possible result of this kind by examining the case of n
lines through one point. Note that this shows that the number of singular points is finite.
Hint: Compare f with a suitable derivative of f which will have degree m − 1.
6. Let C f be a curve of degree m which is irreducible and with orders r i of the singular
points P i . Then show

(m − 1)(m − 2) ≥ r i (r i − 1).
Hint: Compare a suitable derivative of f having order r i − 1at P i with the [(m − 1)(m +
2)/2]-dimensional family of all curves of degree m − 1 and the subfamily of curves
having degree r i − 1at P i . Also, show that this is the best possible result of this kind by
n
examining the curve given by the equation X + WY n−1 = 0.
7. Prove that in characteristic zero a conic C f is reducible if and only if the 3 by 3 matrix of
second partial derivatives of f has a determinant equal to zero. When does this assertion
hold in characteristic p?
8. For a plane curve C f and w 0 : x 0 : y 0 on C f derive the equation of the tangent line
in terms of the first partial derivatives of f at w 0 : x 0 : y 0 . In characteristic zero prove
that w 0 : x 0 : y 0 is a flex if and only if the linear form of first partial derivatives di-
vides the quadratic form of second partial derivatives. When does this assertion hold in
characteristic p?
9. Prove that in characteristic zero the flexes of C f are the intersections between C f and
the curve whose equation is the determinant of the matrix of second partial derivatives of
f . When does this assertion hold in characteristic p?
10. Prove that every nonsingular curve of degree 3 or more has at least one flex over an
algebraically closed field.