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§1. Reduction mod p of Projective Space and Curves 105

(1.3) Remark. Let F(y 0 ,... , y n ) ∈ k[y 0 ,... , y n ] and multiply the polynomial by
an appropriate nonzero element of k such that the coefficients of the new polynomial,
also denoted f , are all in R and have no common irreducible factor. Then we denote
¯
by f (y 0 ,... , y n ) the polynomial over k(p) which is f with all its coefficients re-
¯
duced modulo p. Observe that f is nonzero and homogeneous of the same degree
as f .
(1.4) Definition. Let C be an algebraic curve of degree d in P 2 defined over k.
Choose an equation f = 0 for C of degree d over R with coefficients not having a
common irreducible factor. The reduction mod p of C = C f is the plane curve C ¯ f
of degree d in P 2 defined over k(p).

The modulo p reduction function r p : P 2 (k) → P 2 (k(p)) restricts to a function
(k(p)), for if (w, x, y) ∈ C f (k), then f (w, x, y) = 0, and we can
r p : C f (k) → C ¯ f
apply r p to obtain
¯
¯
0 = r p ( f (w, x, y)) = f ( ¯w, ¯x, ¯y) = f (r p (w, x, y)).
Observe that the same construction and definition holds for hypersurfaces in n vari-
ables over k.
2
(1.5) Examples. The nonsingular conic defined by wx + py = 0 reduces to the
singular conic equal to the union of two lines defined by wx = 0, and the conic
2
2
defined by pwx + y = 0 reduces to y = 0 which is a double line. This exam-
ple shows that there are some subtleties related to reduction modulo p concerning
2
whether y = 0 defines a line or a conic. These questions are taken care of in the
context of schemes, but we do not have to get into this now since a cubic in normal
form reduces to another cubic in normal form.
In order to see what happens with the group law of a cubic under reduction re-
call from 2(4.4) that the intersection multiplicity i(P; L, C f ) of P on L and C f is
defined by forming the polynomial

ϕ(t) = f (w + tw , x + tx , y + ty ),

where P = (w, x, y) and (w , x , y ) ∈ L − C f . The intersection points L ∩ C f are

of the form (w + tw , x + tx , y + ty ) where ϕ(t) = 0, and the order of the zero is

the intersection multiplicity. Further the order of P on C f is always ≤ i(P; L, C f ).
Now we reduce all of the above constructions including the extra polynomial
ϕ(t) mod p from k to k(p). We obtain

¯
¯ ϕ(t) = f ( ¯w + t ¯w , ¯x + t ¯x , ¯y + t ¯y ),

where now we must use further care in choosing (w , x , y ) such that ( ¯w , ¯x , ¯y ) ∈
¯ ¯
L − C ¯ f . This is possible provided L ⊂ C ¯ f .
(1.6) Remark. With the above notations we have the following inequalities:
¯ ¯
i(P; L, C f ) ≤ i(P; L, C f );
¯
order of P on C f ≤ order of P on C ¯ f .