Algebraic Curves/Fall 2015
Aaron Bertram
5. Non-singular and Nodal Curves. Some non-singular and nodal
curves in the plane and other rational surfaces, blowing up the nodes.
Definition 5.1. A plane curve C ⊂ CP2 is nodal if every singular point
of C has a quadratic tangent cone consisting of two distinct lines.
In other words, if C ⊂ CP2 is defined by a prime homogeneous
polynomial F (not divisible by x0 ), the Taylor series of f (y1 , y2 ) = F/xd0
near a point p = (1, b1 , b2 ) ∈ C either has the form:
f (y1 , y2 ) = a1 (y1 − b1 ) + a2 (y2 − b2 ) + higher order terms
with a1 or a2 nonzero, and p ∈ C is a non-singular point or else:
f (y1 , y2 ) = a1 (y1 −b1 )2 +a2 (y1 −b1 )(y2 −b2 )+a3 (y2 −b2 )2 + higher order
with a22 − 4a1 a3 6= 0, in which case p ∈ C is a node.
Example 5.1. (a) A nodal conic is either non-singular else it has
exactly one node and consists of two intersecting lines.
(b) The curve E0 is an irreducible nodal cubic curve with one node.
(c) A cubic divisor with two nodes is the union of a line and a conic.
This is a consequence of Bézout’s Theorem since the line through the
nodes would otherwise have intersection number ≥ 4 with the divisor.
Proposition 5.1. If C ⊂ CP2 is an irreducible nodal
plane curve of
d−1
degree d, then the number of nodes is at most 2 .
Proof. Let p1 , ..., pn ∈ C be the nodes, and let q1 , ..., qr ∈ C be r
additional (nonsingular) points of C, with:
(d − 1)(d + 2)
= dim P(d − 1)
2
Then dim P(d−1; p1 , ..., pn , q1 , ..., qr ) ≥ 0, so there is a divisor D ⊂ CP2
of degree d − 1 passing through each of the p’s and q’s. Since C is
irreducible and D has smaller degree, there is no way for C and D to
share a common component, and so by Bézout’s Theorem,
n+r =
d(d − 1) ≥ 2n + r =
(d − 1)(d + 2)
+n
2
which gives the desired result.
We will eventually explain this result in terms of the genus. First,
though, let’s consider curves in products of projective spaces:
1
2
Proposition 5.2. (a) The Cartesian product CP1 × CP1 embeds in
CP3 as a hypersurface via the map:
Φ((a0 : a1 ), (b0 : b1 )) = (a0 b0 : a0 b1 : a1 b0 : a1 b1 )
so that the image of Φ is the quadric Q := V (x0 x3 − x1 x2 ) ⊂ CP3 .
(b) The two projections πVi : Q → CP1 for:
V1 = {x0 = x2 = 0} ⊂ C4 and V2 = {x0 = x1 = 0} ⊂ C4
coincide with the Cartesian projection maps: πi : CP1 × CP1 → CP1
(and are well-defined on all points of Q).
(c) If F (v0 , v1 ; w0 , w1 ) is bihomogeneous of bidegree (d, e), i.e. if
F (λv0 , λv1 ; µw0 , µw1 ) = λd µe F (v0 , v1 ; w0 , w1 )
then the curve in CP1 × CP1 defined by:
C = {((a0 : a1 ), (b0 : b1 )) | F (a0 , a1 , b0 , b1 ) = 0} ⊂ Q ⊂ CP3
is the intersection of Q with other projective hypersurfaces in CP3 .
(d) Two divisors D and D0 on CP1 × CP1 defined by homogeneous
polynomials of bidegrees (d, e) and (d0 , e0 ) intersect in de0 + d0 e points
(with multiplicities counted with intersection numbers).
Proof. x0 x3 = x1 x2 holds for all points in the image of Φ, and if
c0 c3 − c1 c2 = 0, then
(a0 : a1 ) = (c0 : c2 ) (or (c1 : c3 )) and (b0 : b1 ) = (c0 : c1 ) (or (c2 : c3 ))
inverts the map from CP1 × CP1 to Q and also shows that the two
projections coincide with the Cartesian projections.
We may assume d ≤ e. Then C ⊂ Q ⊂ CP3 is cut out by equations:
0 = x0 x3 − x1 x2 = v0e−d F = v0e−d−1 v1 F = ... = v1e−d F
where v0e−d−i v1i F , of bidegree (e, e), is a homogeneous polynomial of
degree e in the xi via: v0 w0 = x0 , v0 w1 = x1 , v1 w0 = x2 , v1 w1 = x3 .
Finally, consider the bigrading on the polynomial ring:
M
C[v0 , v1 , w0 , w1 ] =
C[v0 , v1 , w0 , w1 ]m,n
m,n≥0
where C[v0 , v1 , w0 , w1 ]m,n are the spaces of bihomogeneous polynomials
of bidegree (m, n), and consider the associated Hilbert polynomials:
h(m, n) = hCP1 ×CP1 (m, n) = dim(C[v0 , v1 , w0 , w1 ]m,n ) = (m + 1)(n + 1)
hD (m, n) = h(m, n) − h(m − d, n − e) = dn + em + 1 − (d − 1)(e − 1)
hD∩D0 (m, n) = hD (m, n) − hD (m − d0 , n − e0 ) = de0 + ed0
3
Then the proof of Bézout’s Theorem follows as in the case of CP2
Example 5.2. (a) The curves C of bidegree (1, 0) and (0, 1) are the
fibers of the two projection maps to CP1 . As such, they embed in CP3
as projective lines. For example, applying Proposition 5.1(c) to the
curve {v0 = 0} ⊂ Q ⊂ CP3 gives equations:
0 = x0 x3 − x1 x2 , 0 = v0 w0 = x0 and 0 = v0 w1 = x2
and the quadratic equation is a consequence of the linear equations.
(b) The curves C of bidegree (1, 1) are intersections of Q with a
hyperplane. Viewed the other way around, the equation for Q becomes
an equation for a conic in CP2 . For example, the equation v0 w0 = v1 w1
becomes: 0 = x0 x3 − x1 x2 and 0 = v0 w0 − v1 w1 = x0 − x3 allowing us
to solve for x3 in terms of x0 , and obtain: C = {x20 − x1 x2 = 0} ⊂ CP2 .
(c) The twisted cubic has bidegree (1, 2) in Q. The equations:
0 = x0 x3 − x1 x2 = x0 x2 − x21 = x22 − x1 x3
are obtained from the single bihomogeneous equation: 0 = v1 w02 −v0 w12 .
(d) Let C be defined by an equation of bidegree (2, g + 1):
0 = v02 G0 (w0 , w1 ) + v0 v1 G1 (w0 , w1 ) + v12 G2 (w0 , w1 ) = F (v0 , v1 , w0 , w1 )
where the Gi are homogeneous of degree g + 1. The discriminant:
∆ = G21 − 4G0 G2
is a (homogeneous) polynomial of degree 2g + 2.
Over a point (w0 : w1 ) = b ∈ CP1 with ∆(b) 6= 0, there are two nonsingular points of C projecting to b or else the intersection C ∩ π2−1 (b)
is one point with local intersection number two (by Bézout). Above a
point satisfying ∆(b) = 0, there are two possibilities:
Either the zero is a simple zero of the polynomial ∆, and the curve C
is nonsingular at the point above b, with Zariski tangent line π2−1 (b), or
else the zero has higher multiplicity and the point over b is a singular
point of C with non-vertical tangent lines. We can break this down
further. If the zero is of multiplicity two, the point over b is a node, and
if the multiplicity is greater than two, the two tangent lines coincide.
Thus:
(i) If every root of ∆ is simple, then C is non-singular, and
(ii) If every root of ∆ has multiplicity ≤ 2, then C is a nodal.
Finally, note that the nonsingular curves C of type (2, g + 1) are all
hyperelliptic via the projection π2 .
Exercise 5.1. Prove (i) and (ii) above carefully in affine coordinates.
4
Our next aim is to prove the following:
Theorem 5.1. Every nodal curve C ⊂ CP2 is the image of a projection
e → CP2 from a non-singular curve C
e ⊂ CPn mapping bijectively
π:C
to all nonsingular points of the image and 2 : 1 onto each node.
e “separates the branches” of each of the nodes of C.
i.e.. the curve C
It’s time for a definition of a non-singular curve in CPn :
Definition 5.2. A (connected) intersection of hypersurfaces:
C = V (F1 ) ∩ · · · ∩ V (Fm ) ⊂ CPn
is a non-singular embedded curve if, for every point p ∈ C, some choice
of n − 1 of the hypersurfaces are individually non-singular at p with
transverse Zariski tangent spaces and cut out C near p.
Example 5.3. (a) A nonsingular curve in CP2 with one equation.
(b) The twisted cubic C3 ⊂ CP3 expressed as:
C3 = V (x0 x3 − x1 x2 ) ∩ V (x0 x2 − x21 ) ∩ V (x1 x3 − x22 )
For each p ∈ C3 , one of the pairs of quadrics will cut out C3 near p
with transverse Zariski tangent spaces, but the choice of the pair
depends upon the point. At (1 : 0 : 0 : 0), the first two quadrics will
work, but near (0 : 0 : 0 : 1), the first and third quadrics are required.
In fact, every pair of quadric hypersurfaces vanishing on C3 cut out
an additional line, so three is the minimum necessary to define C3 .
Exercise 5.2. (a) Suppose F ∈ C[v0 , v1 , w0 , w1 ]d,e is prime and:
∇F ((a0 : a1 ), (b0 : b1 )) 6= (0, 0, 0, 0)
Check in affine coordinates that this is equivalent to the non-vanishing
of the linear term in the Taylor series for F near p ∈ C ⊂ CP1 × CP1 .
(b) Show also that this condition guarantees that the intersection of
the Zariski tangent spaces to the hypersurfaces from Proposition 5.2(c)
at p ∈ C ⊂ Q ⊂ CP3 is one-dimensional, so we may view C = V (F ) as
being non-singular both as a curve in CP1 × CP1 and in CP3 .
The idea behind the proof of Theorem 5.1 is to “blow up” the nodes.
We will do this and locate the desired curve inside:
CP2 × CP1 × · · · × CP1 ⊂ CPn
where the number of CP1 factors is the same as the number of nodes.
The value of n will be large. If there are m nodes, then n = 3 · 2m − 1
which will be shown later to be rather far from optimal.
5
Blowing Up One Point. Consider the graph of the projection:
γ : CP2 − − > CP2 × CP1 ; (a0 : a1 : a2 ) 7→ ((a0 : a1 : a2 ), (a1 : a2 ))
from the point p = (1 : 0 : 0). Then:
(i) The bihomogeneous polynomial of bidegree (1, 1):
F = x1 y2 − x2 y1 ∈ C[x0 , x1 , x2 , y1 , y2 ]
vanishes on the image of γ and the “horizontal” line over p:
S := V (F ) = γ(CP2 − {p}) ∪ ({p} × CP1 ) ⊂ CP2 × CP1
and as in Example 5.2(b), S is the intersection of the embedded:
CP2 × CP1 ⊂ CP5 ; ((a0 : a1 : a2 ), (b1 : b2 )) 7→ (... : ai bj : ...)
with a single hyperplane.
(ii) Let C = V (G) ⊂ CP2 be an irreducible plane curve of degree d
passing through p. Then the intersection:
S ∩ V (G(x0 , x1 , x2 )) ⊂ CP2 × CP1
consists of two “component curves,” the image of C under γ and the
horizontal line. To see where the two components should intersect:
(iii) If C is non-singular at p, then the image of C under γ is also
non-singular. On the other hand, if C has a node at p, then:
2
2
d−3
G = xd−2
0 (ax1 + bx1 x2 + cx2 ) + x0 G3 (x1 , x2 ) + · · · + Gd (x1 , x2 )
and we may use the relation x1 y2 = x2 y1 to set each x2 = x1 (y2 /y1 ),
substitute into G and clear the denominator. This gives us a new:
2
2
d−3
G0 = xd−2
0 (ay1 + by1 y2 + cy2 ) + x0 G3 (x1 , x2 , y1 , y2 ) + ...
of bidegree (d − 2, 2) that (together with x1 y2 = y1 x2 ) cuts out a
curve B consisting of γ(C) and two additional points on p × CP1
corresponding to the two tangent lines to C at p, interpreted as points
of CP1 . These points are non-singular on the curve B, and, via the
projection to CP2 , the goal of Theorem 5.1 has been achieved above p.
Blowing Up Many Points: The procedure above can be performed
with many projections maps at once:
γ : CP2 − − > CP2 × CP1 × ... × CP1
as the simultaneous graph of n projection maps (πp1 , ..., πpn ) from
points p1 , ..., pn ∈ CP2 . Each contributes curves Bi associated to equations G0i . The intersection of these cuts out a single curve B that
separates the nodes, and embeds in CPn via the “bidgree (1, ..., 1)”
embedding of the CP2 × CP1 × · · · × CP1 .