Algebraic Curves/Fall 2015 Aaron Bertram

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Algebraic Curves/Fall 2015
Aaron Bertram
8. Linear Series. Some applications of the Riemann-Roch Theorem.
Let D ≥ 0 be a divisor on a non-singular curve C.
Definition 8.1. The linear series |D| is the projective space of effective
divisors that are linearly equivalent to D, i.e.
|D| = {E ≥ 0 | ∃φ ∈ K(C) such that div(φ) + D = E}
This is the projective space of lines through the origin in L(D).
The dual projective space |D|∨ is the projective space of codimension
one hyperplanes in |D| (or subspaces V ⊂ L(D) of codimension one).
This may alternatively be understood as the projective space of lines
through the origin in the dual vector space L(D)∨ .
Definition 8.2. A point p ∈ C is a base point of |D| if p ∈ E for all
E ∈ |D|, or equivalently, if L(D − p) = L(D).
Suppose that |D| is a base-point-free linear series. Then:
Proposition 8.1. The (instrinsically defined) linear series mapping:
Φ|D| : C → |D|∨ ; Φ|D| (p) = L(D − p)
may be realized by choosing a basis φ0 , ..., φr(D) for L(D) and setting:
Φ(p) = (φ0 (p) : ... : φr(D) (p)) ∈ CPr(D)
Proof: The first map is well-defined by definition. The second map
is a priori only defined for p 6∈ Supp(D), since by definition of L(D),
the φi are all defined P
at p and some φ(p) 6= 0. But in fact Φ is defined
everywhere. If D =
di pi , then let ypi ∈ mC,pi be a generator, and
notice that:
((ypdii φ0 )(p) : ... : (ypdii φr(D) )(p)) = Φ(p)
is defined at pi and agrees with the other expression whenever both are
defined, again because pi ∈ C is not a base point of L(D).
∨
Consider the basis x0 , ...,
Pxr(D) ∈ L(D) ∨ dual to φ0 , ..., φr(d) ∈ L(D).
A point (aP
ai xi ∈ L(D) with respect to this basis
0 , ..., ar(D) ) =
evaluates ( ai xi )(φi ) = ai . Thus if p 6∈ Supp(D), then the vector:
X
(φ0 (p), ..., φr(D) (p)) =
φi (p)xi ∈ L(D)∨
P
satisfies ( φi (p)xi )(φi ) = φi (p), i.e. it is the linear map from L(D) to
C given by evaluation at p, which has kernel L(D − p). One reasons
similarly for each of the pi ∈ Supp(D).
1
2
The latter description shows Φ is defined by rational functions, with
the hyperplanes in |D|∨ in natural correspondence (via intersecting C)
with the divisors E ∈ |D|. In particular, the image of C spans |D|∨ in
the sense that it is not contained in any hyperplane.
Corollary 8.1. If |D| and |D − p| are base-point-free ∀p ∈ C , i.e. if
l(D − p − q) = l(D) − 2
for all p, q ∈ C (including p = q), then Φ|D| is an embedding.
Proof. If p 6= q, then since L(D − p − q) = L(D − p) ∩ L(D − q)
and l(D − p − q) = l(D − p) − 1 (by assumption), it follows that
L(D − p) 6= L(D − q), so the linear series map is injective. Moreover,
L(D − p − q) defines the secant line through Φ(p) and Φ(q) via:
L(D − p − q) ⊂ L(D) ↔ CP1 = P((L(D)/L(D − p − q))∨ ) ⊂ P(L(D)∨ )
Similarly, L(D − 2p) defines the Zariski tangent line at Φ(p) via:
L(D − 2p) ⊂ L(D) ↔ CP1 = P((L(D)/L(D − 2p))∨ ) ⊂ P(L(D)∨ )
from which we conclude that Φ(C) ⊂ |D|∨ is a non-singular curve. We now meet every non-singular curve, courtesy of Riemann-Roch:
Proposition 8.2. Every non-singular curve C of genus g ≥ 2 either:
(i) embeds in CPg−1 = |KC |∨ via the canonical linear series, or else:
(ii) embeds in CP1 ×CP1 as a non-singular curve of bidegree (2, g+1).
The latter are the hyperelliptic curves of genus g ≥ 2.
Proof. By Riemann-Roch, l(KC )=g and KC is base-point-free since:
l(KC − p) − l(p) = (2g − 3) + 1 − g = g − 2
and l(p) = 1 for all curves other than CP1 (as noted in §1). Similarly,
l(KC − p − q) − l(p + q) = (2g − 4) + 1 − g = g − 3
and if l(p + q) > 1, then there is a non-constant φ ∈ K(C) defining:
φ : C → CP1 of degree 2
P
and the computation deg(dφ) = 2g −2 = −4+ p∈C (ep −1) shows that
φ has 2g + 2 branch points lying over points r1 , ..., r2g+2 ∈ CP1 . The
curve C is determined by the branching, and since the branch points
can be arranged to be the zeroes of the discriminant of a suitable:
F (v0 , v1 , w0 , w1 ) ∈ C[v0 , v1 , w0 , w1 ]2,g+1
as in Example 5.2(d), it follows that C = V (F ) ⊂ CP1 × CP1 .
Otherwise l(p + q) = 1 for all p, q and Φ|KC | is an embedding.
3
Note that every non-singular curve of genus 2 is hyperelliptic, since:
deg(KC ) = 2g − 2 = 2 and l(KC ) = g = 2
To find non-hyperelliptic curves of higher genus, we need a criterion
for determining when a curve C ⊂ CPn is canonically embedded.
Proposition 8.3. If D is a divisor of degree 0, then:
l(D) = 0 unless D = div(φ), for φ ∈ K(C)∗ , in which case l(D) = 1
and if D0 is a divisor of degree 2g − 2, then:
l(D0 ) = g−1 unless D0 = div(ω), for ω ∈ Ω(C), in which case l(D) = g
Proof. For the first statement if l(D) > 0, then φ ∈ L(D) satisfies
div(φ) + D = E ≥ 0. But deg(E) = 0, so E = 0 and D = div(φ−1 ).
The second is then a consequence of the first and Riemann-Roch, since:
l(D0 ) − l(KC − D0 ) = g − 1 and deg(KC − D0 ) = 0
Corollary 8.2. If C ⊂ CPg−1 has genus g and spans CPg−1 , then:
(i) deg(C) ≥ 2g − 2
(ii) If deg(C) = 2g − 2, then the embedding is canonical.
In particular, in case (ii), C is not hyperelliptic.
Proof. Since C spans CPg−1 , we have R1 = hx0 , ..., xg−1 i where R
is the homogeneous coordinate ring of C. Let Hi = div(xi ) for any of
the coordinate functions. Then the injection f1 : R1 → L(Hi ) gives
l(Hi ) ≥ g. If deg(C) = deg(Hi ) ≤ 2g − 2, then:
l(Hi ) − l(KC − Hi ) = deg(Hi ) + 1 − g ≤ g − 1
so l(KC − Hi ) > 0, and there is an effective divisor E ∈ |KC − Hi | so
that E + Hi = KC , which in particular implies that l(Hi ) ≤ g = l(KC ),
with equality if and only if E = 0 and the embedding is canonical. Examples of Canonically Embedded Curves:
(3) C = V (F4 ) ⊂ CP2 , a non-singular quartic plane curve.
(4) C = V (F2 ) ∩ V (G3 ) ⊂ CP3 for a quadric and cubic.
(5) C = V (F2 ) ∩ V (G2 ) ∩ V (H2 ) ⊂ CP4 for three quadrics.
To see each of these, it suffices to compute their Hilbert polynomials
(to determine their degree and genus), which are, respectively:
(3) 4d + (1 − 3) giving degree 4 and genus 3.
(3) 6d + (1 − 4) giving degree 6 and genus 4.
(4) 8d + (1 − 5) giving degree 8 and genus 5.
4
Proposition 8.4. The converse holds for Examples (3) and (4).
Proof. Let C be a non-singular non-hyperelliptic curve of genus 3
and consider the canonical embedding:
Φ : C → |KC |∨ = CP2
The Hilbert polynomials of C and of a plane quartic are the same,
suggesting that C is a non-singular quartic. To see that this is, in fact,
the case, consider the mapping f4 : R4 → L(4KC ).
Since R4 = (C[x0 , x1 , x2 ]/I(C))4 has dimension 15−dim(I(C)4 ), and
by Riemann-Roch:
l(4KC ) = 8g − 8 + 1 − g = 7g − 7 = 14
it follows that C does, in fact, lie on a quartic, which then must be
equal to C since their Hilbert polynomials agree.
If C has genus 4, we reason similarly:
f2 : C[x0 , .., x3 ]2 /I(C)2 → L(2KC )
has (at least) a one-dimensional kernel, since:
dim(C[x0 , ..., x3 ])2 = 10 and l(2KC ) = 3g − 3 = 9
but C ⊂ |KC |∨ = CP3 cannot lie in the intersection of two (distinct)
quadrics since such an intersection is either a plane or else has a linear
Hilbert polynomial with leading term 4d, and so cannot contain C,
which has Hilbert polynomial 6d − 3. Let V (F2 ) be the quadric. Next,
f3 : C[x0 , ..., x3 ]3 → L(3KC )
has a kernel of dimension (at least) 5, implying that C ⊂ |KC |∨ lies on
one additional cubic outside the span of the four linearly independent
cubics x0 F2 , ..., x3 F2 . The Hilbert polynomial of C[x0 , ..., x3 ]/hF2 , G2 i
is 6d + (1 − 4), from which we may conclude that C = V (F2 ) ∩ V (G2 )
(and that f3 is surjective).
When we apply this reasoning to C of genus 5, however, we get:
f2 : C[x0 , ..., x4 ]2 → L(2KC ) of dimensions 15 and 12, respectively
which does tell us that C ⊂ CP4 lies on (at least) 3 linearly independent
quadrics, however, it may be the case that:
V (F2 ) ∩ V (G2 ) ∩ V (H2 ) = S ⊂ CP4
is a surface. This does happen if C admits a 3 : 1 map φ : C → CP1 ,
as we shall see.
5
Let Φ : C → |D|∨ = CPr be the linear series mapping
associated
P
to a base-point-free linear series |D|, and let E =
ei pi ≥ 0 be an
effective divisor on C. Then:
Definition 8.3. The linear span of E in CPr is the intersection of all
the hyperplanes H ⊂ CPr with the property that:
div(H) − E ≥ 0
Remark. If each ei = 1 and Φ(pi ) 6= Φ(pj ), then the linear span agrees
with the “ordinary” linear span of the points Φ(pi ), i.e. it is the smallest
(projective) subspace CP(V ) ⊂ CPr containing each Φ(pi ). The points
are linearly independent if and only if the dimension is deg(E).
Proposition 8.5. In the setting above, the (projective) dimension of
the linear span of E is r − l(D − E).
Proof. By definition, l(D−E) is the number of linearly independent
φ ∈ L(D) that vanish along the divisor E.
Corollary 8.3. The linear span of an effective divisor E under the
canonical embedding C ⊂ CPg−1 of a non-hyperelliptic curve C has
(projective) dimension:
deg(E) − l(E)
e.g. if each ei = 1, then l(E) − 1 measures the “amount” by which the
points pi fail to be linearly independent under the canonical embedding.
Proof. By Riemann-Roch and Proposition 8.5, the dimension is:
(g − 1) − l(KC − E) = deg(E) − l(E) Example. Consider a non-singular genus 6 quintic curve:
C = V (F5 ) ⊂ CP2
composed with the Veronese embedding CP2 ⊂ CP5 given by:
(s0 : s1 : s2 ) 7→ (s20 : s0 s1 : s21 : s0 s2 : s1 s2 : s22 )
P
The intersection of C ⊂ CP2 ⊂ CP5 with the hyperplane
ai x i = 0
is the intersection of C ⊂ CP2 with the conic a0 s20 + ... + a5 s22 = 0,
which has degree 10 (by Bézout’s Theorem), so by the earlier criterion,
we see that C ⊂ CP5 is a canonically embedded curve of genus 6.
Consider the span (in CP5 ) of the 5 points of C ∩ L ⊂ CP2 ⊂ CP5 ,
for a line L ⊂ CP2 . Then under the Veronese embedding,
p1 , ..., p5 ∈ L ⊂ CP5
the image of L is a conic in a linear CP2 = P(V ) ⊂ CP5 spanned by
Φ(p1 ), ..., Φ(p5 ), so the five points fail by two (projective) dimensions
to be linearly independent, and indeed l(p1 + · · · + p5 ) = 3.
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Proposition 8.6. Let C ⊂ CPg−1 be the canonical embedding. If:
(i) C admits a 3 : 1 map to CP1 , or
(ii) C ⊂ CP2 is a non-singular plane quintic,
then the intersection of the quadrics containing C contains a surface.
Proof. If C admits a 3 : 1 map φ : C → CP1 , then each inverse
image φ−1 (x) = Dx is a divisor on C of degree three. By the Corollary,
since l(Dx ) = 1, the image Dx ⊂ CPg−1 spans a projective line, which
is a trisecant line to the canonically embedded curve. Each such line
CP1 = Dx ⊂ CPg−1 is contained in every quadric that contains C
(since the restriction of the equation of the quadric is either zero or else
has at most two zeroes). Thus the quadrics containing C also contain
a family of lines in CPg−1 parametrized by the points of |Dx | ∼
= CP1 .
Similarly, if C = V (F5 ) ⊂ CP2 , then as in the example above, if
we let D = p1 + · · · + p5 , then all the divisors Dx ∈ |D| lie on conics
(the images of L ⊂ CP5 , and any quadric in CP5 vanishing on C must
contain the conic. Notice that in this case, the union of these conics
is the the union of the images of lines L ⊂ CP2 , which is the image of
CP2 itself under the Veronese embedding.
This analysis of examples begs several questions:
Definition 8.4. A non-singular curve C ⊂ CPr = |D|∨ is projectively
normal if all of the maps:
fk : C[x0 , ..., xr ]d → L(kD)
are surjective.
Remarks. The map f1 is surjective if and only if C spans CPr , and we
saw in Proposition 7.1 that the fk are surjective for sufficiently large
k. Notice that if a curve C ⊂ CPr is projectively normal, then:
k+r
dim(I(C)k ) =
− l(kD) for all k ≥ 0
r
since I(C)k is the kernel of fk .
Question 1. Is the canonically embedded curve projectively normal?
Question 2. Under what circumstances is the canonically embedded
curve not an intersection of quadrics, and in those cases, what is the
intersection of the quadrics containing C?
The answers to these questions have to do with the Clifford Index of
the curve, which we introduce now.
7
Definition 8.5. A divisor D on C is special if:
l(D) > 0 and l(KC − D) > 0
Clifford’s Inequality. If D is a special divisor on C, then:
deg(D)
l(D) ≤
+1
2
Proof. This follows the inequality:
(∗) l(D + D0 ) ≥ l(D) + l(D0 ) − 1
(for any pair of (effective) divisors D, D0 on C), since it gives:
g = l(KC ) ≥ l(D) + l(KC − D) − 1
and then by the Riemann-Roch Theorem:
l(KC − D) = l(D) − deg(D) + g − 1
and Clifford’s inequality immediately follows.
Notice that the inequality (∗) is equvialent to:
dim(|D + D0 |) ≤ dim(|D|) + dim(|D0 |)
but this follows from dimension considerations, since the sum mapping:
s : |D| × |D0 | → |D + D0 |; (E, E 0 ) 7→ E + E 0
is “quasi-finite,” i.e. each element E 00 ∈ |D + D0 | has only a finite
number of preimages for the map s, corresponding to subsets of E 00
that are in |D|. Moreover, since the image of the sum mapping is a
closed irreducible subvariety of |D + D0 |, it also follows that:
(**) if l(D + D0 ) = l(D) + l(D0 ) − 1, then s is surjective.
This will be useful later.
Examples. The divisors 0 and KC both satisfy the equality:
deg(D)
l(D) =
+1
2
and C is hyperelliptic if C has a divisor D such that:
deg(D) = 2 and l(D) = 2
which also satisfies the equality. We will see in §9 that only hyperelliptic curves admit special divisors other than 0 and KC for which the
Clifford inequality is an equality.
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