Segre embeddings and the canonical image of a curve

0.12 MB·English

by  Nathan Grieve

SEGRE EMBEDDINGS AND THE CANONICAL IMAGE OF A CURVE
NATHANGRIEVE
2
1
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n ABSTRACT. We prove that there is no g for which the canonical embedding of a general
a
curve of genus g lies on the Segre embedding of any productof three or more projective
J
spaces.
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] 1. INTRODUCTION
G
Ifg iscompositethenthecanonicalembeddingofageneralcurveofgenusg liesonthe
A
Segre embedding of a product of two projective spaces. For example, a general curve of
.
h genus 4 lies on the Segre embedding of P1 ×P1 while a general curve of genus 6 lies on
t
a the Segre embedding of P1 ×P2. These facts have applications concerning the structure
m
Chow ring of M as illustrated in [Fab, p. 421] and [Pen, p. 26]. The aim of this note is
g
[
to prove that, by contrast, there is no g for which the canonical embedding of a general

1 curve of genus g lies on the Segre embedding of any product of three or more projective
v
6 spaces.
8 To prove our main result we first give the following criterion for a general curve to lie
0 on some Segre embeddingPr1 ×···×Prn → Pg−1.
2
.
1 Proposition 1.1. The canonical image of a general curve C of genus g lies on some Segre em-
0 bedding Pr1 × ··· × Prn → Pg−1 if and only if C admits line bundles L ,...,L such that
2 1 n
1 n n
L = K , h0(C,L ) = r +1 and (r +1) = g.
: i C i i i
v
i=1 i=1
i
X N Q
Wethen prove the following stronger result ofindependentinterest.
r
a
Theorem 1.2. Let C be a general curve of genus g and let n > 3. If C admits line bundles
n
L ,...,L suchthat degL = 2g−2andh0(C,L ) = r +1 > 2fori = 1,...,n,thenn = 3
1 n i i i
i=1
and P
3 3
r +r +r +2
(r +1) < (r +1) 1 2 3 6 g.
i i
r +r +r +2−r r r
i=1 (cid:18)i=1 (cid:19)(cid:18) 1 2 3 1 2 3(cid:19)
Y Y
Moreover,up topermutationof the r , wehave
i
(a) r = r = 1 and 1 6 r 6 g −2or
1 2 3 4
(b) r = 1,r = 2and r = 2,3,or 4.
1 2 3
MathematicsSubjectClassification(2010):14H10.
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2 NATHANGRIEVE
Amore conceptual interpretation ofthe theorem isthe following corollary.
Corollary 1.3. LetC bea generalcurveof genus g.
(a) If g iscompositethenthe canonical imageof C liesonsome Segreembedding
Pr1 ×Pr2 → Pg−1.
(b) If n > 3 thenthecanonical imageof C doesnot lieonany Segreembedding
Pr1 ×···×Prn → Pg−1.
Statement (a) in Corollary 1.3 is well-known to experts and follows from Brill-Noether
theory. We include a proof for completeness. We refer to [ACGH, Chap. IV] and [ACG,
Chap. XXI] for the basic results in Brill-Noether theory and [ACGH, p. xv] for notation.
Forthenon-expertwegiveaquickexposition ofsomeofthemainresultsofthetheoryin
Section 2. Throughout we assume thatg is aninteger greater than or equalto 3.
Acknowledgments. I would like to thank my PhD advisor Mike Roth for several useful
discussions, for encouraging me to write this note and for reading a preliminary draft.
I also benifited from conversations with Greg Smith and was partially supported by an
Ontario Graduate Scholarship.
2. BRILL-NOETHER THEORY
For all smooth curves C of genus g there exist moduli schemes Wr(C) whose closed
d
points consist of equivalence classes of degree d line bundles on C with at least r + 1
global sections. Explicitly the closed points of Wr(C) are identified with the set
d
{L | L ∈ Pic(C),degL = d and h0(C,L) > r +1}.
For the construction of Wr(C) see[FL, p. 279]or [ACGH, p. 176].
d
Brill-Noether theory studies the geometry of such Wr(C). Central to the theory is the
d
Brill-Noether numberandthe Petri-map. TheBrill-Noether numberisdefined by
ρ(g,r,d) := g −(r +1)(g +r −d).
See [ACGH, p. 159] for an explanation as to how this number arises. On the other hand,
the Petri-map is defined for all line bundles L on C. If K denotes the canonical bundle
C
ofC then the Petri-map isthe cup-product
µ(L) : H0(C,L)⊗H0(C,L∨ ⊗K ) → H0(C,K ).
C C
Both the Brill-Noether number and the Petri-map have remarkable geometric impli-
cations as seen in the two main theorems of Brill-Noether theory which we now de-
scribe. The first theorem appliesto all smooth curves and isthe result of work byKempf,
Kleiman-Laksov andFulton-Lazarsfeld.
Theorem 2.1 (Brill-Noether theory theorem I [KL], [FL]). Let C be a smooth curve of genus
g.
SEGREEMBEDDINGSANDTHECANONICALIMAGEOFACURVE 3
(a) If ρ(g,r,d) > 0 then Wr(C) 6= ∅ and every irreducible component of Wr(C) has dimen-
d d
siongreaterthan or equal toρ(g,r,d).
(b) If ρ(g,r,d) > 1 thenWr(C)is connected.
d
If C is a general curve then the converse to Theorem 2.1 (a) holds – we state this in
the second main theorem (Theorem 2.4). That the converse to Theorem 2.1 (a) holds for
general curves wasfirst proved byGriffiths-Harris [GH]usinga degeneration argument.
Petriconjectured thatµ(L)isinjective foralllinebundlesLonageneralcurveC. (This
is implicit in [Pet]. See the footnote on [ACGH, p. 215] for a discussion.) Arbarello-
Cornalba clarified the geometric implications of thisconjecture.
Theorem 2.2 (Arbarello-Cornalba theorem [AC, Theorem 0.3]). Let C be a general curve of
genus g. If µ(L) is injective for all line bundles L on C and Wr(C) 6= ∅ then every irreducible
d
componentof Wr(C) has dimensionρ(g,r,d)and Wr(C)is non-singular away fromWr+1(C).
d d d
In [Gie] Gieseker used a degeneration argument, and which was subsequently stream-
lined by Eisenbud-Harris [EH], to prove that µ(L) is injective for all line bundles on a
general curve. Lazarsfeld, without using degenerations, also gave an independent proof
[Laz, p. 299]. The fact that µ(L) is injective for all line bundles on a general curve is
sometimes referred to asthe Gieseker-Petri theorem.
Theorem 2.3 (Gieseker-Petri theorem [Gie, Theorem 1.1, p. 251]). If C is a general curve of
genus g thenthe cup-productµ(L) isinjectivefor allline bundles LonC.
The above discussion, combined with [FL, Corollary 2.4, p. 280], is summarized in the
second main theorem ofBrill-Noether theory.
Theorem 2.4 (Brill-Noether theory theorem II [AC], [FL], [Gie],[GH], [Laz]). Let C be a
generalcurveof genus g.
(a) If Wr(C) 6= ∅ then ρ(g,r,d) > 0, Wr(C) is of pure dimension ρ(g,r,d) and Wr(C) is
d d d
non-singular away fromWr+1(C).
d
(b) If ρ(g,r,d) > 1 thenWr(C)is irreducible.
d
Theresults ofthis note dependon Theorem 2.1(a), Theorem 2.3, andTheorem 2.4(a).
3. THE n-FOLD PETRI-MAP AND THE PROOF OF PROPOSITION 1.1
Proposition 1.1 relies on the following lemma which implies that, for a general curve,
the n-fold Petri-map isinjective.
Lemma 3.1. Let C be a general curve of genus g. For all line bundles L ,...,L on C such that
1 n
n n
L ∼= K the cup-product H0(C,L ) → H0(C,K ) is injective.
i C i C
i=1 i=1
N N
Proof. The case n = 1 is trivial. The case n = 2 is the Gieseker-Petri theorem (Theorem
2.3). Letn > 3. Without lossof generality wemayassume that H0(C,L ) 6= 0, i = 1,...n.
i
4 NATHANGRIEVE
Thecup-product factors
H0(C,L )⊗···⊗H0(C,L ) // H0(C,K ) .
1 (cid:15)(cid:15) ❣❣❣❣❣n❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣33 C
H0(C,L ⊗L )⊗H0(C,L )⊗···⊗H0(C,L )
1 2 3 n
By induction the diagonal arrow is injective. It thus suffices to show that the downward
arrow isinjective. For thiswe reduce to showing that the cup-product
Φ : H0(C,L )⊗H0(C,L ) → H0(C,L ⊗L )
1 2 1 2
isinjective.
By assumption there exist non-zero sections σ ∈ H0(C,L ),i = 3,...n. These produce
i i
(via cup-product) a non-zerosection
σ = σ ···σ ∈ H0(C,L ⊗···⊗L ).
3 n 3 n
Since σ 6= 0 multiplication by σ yieldsthe following two injections
H0(C,L ) → H0(C,L ⊗···⊗L )
2 2 n
and
H0(C,L ⊗L ) → H0(C,K ).
1 2 C
Usingthe above we produce (bycup-product) the commutative diagram
H0(C,L )⊗H0(C,L ) Φ // H0(C,L ⊗L ) .
1 2 1 2
id⊗·σ ·σ
(cid:15)(cid:15) (cid:15)(cid:15)
H0(C,L )⊗H0(C,L ⊗···⊗L ) µ(L1) // H0(C,K )
1 2 n C
The vertical arrows of the above diagram are injective, as just noted, whereas bottom
arrowofthediagramisinjective bytheGieseker-Petri theorem. Hencethetop arrowΦis
injective. (cid:3)
Wenowuse Lemma3.1 toprove Proposition 1.1.
Proof of Proposition1.1. Let η : C → Pg−1 be the canonical map. If C is contained in the
image of some Segre embedding φ : Pr1 × ··· × Prn → Pg−1 then there exists a closed
immersion ψ : C → Pr1 ×···×Prn makingthe diagram
C η // Pg−1
ψ (cid:15)(cid:15) ♦♦♦♦♦♦♦φ♦♦♦♦♦77
Pr1 ×···×Prn
commute.
SEGREEMBEDDINGSANDTHECANONICALIMAGEOFACURVE 5
For every 1 6 i 6 n, let π denote the projection of Pr1 × ···× Prn onto the i-th factor
i
and setLi := (πi ◦ψ)∗OPri(1) ∼= ψ∗(πi∗OPri(1)). Wethen obtain
KC ∼= η∗OPg−1(1) ∼= (φ◦ψ)∗OPg−1(1) ∼= L1 ⊗···⊗Ln.
Since the canonical image of C is non-degenerate we conclude that h0(C,L ) > r +1 for
i i
i = 1,...,n.
Finally, since η is induced by the complete canonical series, we conclude that the cup-
n
product H0(C,L ) → H0(C,K )issurjective. ByLemma3.1thecup-productisinjective
i C
i=1
n
N
and,by assumption, (r +1) = g. Itfollows that h0(C,L ) = r +1for i = 1,...,n.
i i i
i=1
Conversely, given sQuch L1,...,Ln we get regular maps C → Pri for i = 1,...,n. We
thus can make a regular map η : C → Pg−1 (induced by a (sub)-canonical series) by
composition C // Pr1 ×···×Prn Segre// Pg−1 .By Lemma3.1the cup-product
n n
H0(C,L ) → H0(C,K ) is injective. Since (r + 1) = g it is also surjective. We thus
i C i
i=1 i=1
cNonclude that the resulting map η isgiven byQthe complete canonical series. (cid:3)
4. PROOF OF MAIN THEOREM AND ITS COROLLARY
We first prove Corollary 1.3 (a). We then prove Theorem 1.2 from which we deduce
Corollary 1.3 (b).
The case n = 2 and g is composite. When n = 2 and g is composite it is easy to prove
that, in its canonical embedding, a general curve of genus g lies on the image of some
(non-trivial) Segre embeddingPr1 ×Pr2 → Pg−1.
Proof of Corollary 1.3(a). Since g iscomposite wecanwrite g = (r +1)(r +1)with r > 1.
1 2 i
Setd = r1g +r = r r +2r . Thenρ(g,r ,d ) = 0so,byTheorem2.1(a),C admitsaline
1 r1+1 1 1 2 1 1 1
bundle L with at least r + 1 global sections. On the other hand ρ(g,r + 1,d ) < 0, so
1 1 1 1
Theorem 2.4 (a) implies that Wr1+1 = ∅. Hence L has exactly r +1 global sections. Set
d1 1 1
L := L∨ ⊗K . Then, by the Riemann-Roch theorem, we obtain h0(C,L ) = g −d +r .
2 1 C 2 1 1
Simplifying and usingour expressions for g and d above we obtain
1
h0(C,L ) = (r + 1)(r + 1) − r r − 2r + r = r + 1. The assertion now follows from
2 1 2 1 2 1 1 2
Proposition 1.1. (cid:3)
The case n > 3. When n > 3 and r > 1, for i = 1,...,n, the situation is in stark contrast
i
to the case n = 2. Indeed we prove that there is no g for which the canonical embedding
ofageneralcurveliesontheSegre embeddingofanyproduct ofthreeormoreprojective
spaces. We deduce this result from Theorem 1.2 whose proof occupies the rest of this
section. The theorem follows from the following more general observation.
6 NATHANGRIEVE
Proposition 4.1. Let C be a general curve of genus g. Let d be a non-negative integer and let
n
q := d . IfC admitslinebundlesL ,...,L suchthatr +1 = h0(C,L ) > 2anddeg L = d,
g 1 n i i i
i=1
j k n N
1+ ri
thenn < 2q +2 and g > (cid:18)iP=1 (cid:19) .
n
−n+(q+1)+ 1
(cid:18)iP=1ri+1(cid:19)
Proof. Since C is general,ifsuch L exist then, byTheorem 2.4(a),
i
ρ(g,r ,d ) = g −(r +1)(g +r −d ) > 0 for i = 1,...,n.
i i i i i
Solving for d we conclude d > g +r − g for i = 1,...,n. Let r denote the remainder
i i i ri+1
obtained by dividingd byg. Then 0 6 r < g and
n n n
g
d = (q +1)g +r −g = d > ng + r − .
i i
r +1
! i !
i=1 i=1 i=1
X X X
Rearrangingwe obtain
n n
1
(4.1) n−(q +1)− g 6 r −g − r < 0.
i
r +1
i !! !
i=1 i=1
X X
Since g > 0we conclude
n
1
(4.2) n−(q +1)− < 0.
r +1
i !
i=1
X
Nowby assumption r > 1,for i = 1,...,n. Thus
i
n
1 n
(4.3) 6 .
r +1 2
i
i=1
X
Usingequations(4.3)and(4.2)wededucethatn < 2q+2. Finally,dividingequation(4.1)
byequation (4.2)we obtain
n n
−r +g + r 1+ r
i i
(4.4) g > (cid:18)i=1 (cid:19) > (cid:18)i=1 (cid:19) .
Pn P n
−n+(q +1)+ 1 −n+(q +1)+ 1
ri+1 ri+1
(cid:18)i=1 (cid:19) (cid:18)i=1 (cid:19)
P P (cid:3)
Wenowuse Proposition 4.1 to prove Theorem 1.2.
Proof of Theorem1.2. Setd = degL ,i = 1,...,nandsetd = 2g−2. Thenq := d = 1and
i i g
the remainder r equals g − 2. Since ri > 1, for i = 1,...,n, applying Proposjitiokn 4.1 we
SEGREEMBEDDINGSANDTHECANONICALIMAGEOFACURVE 7
concludethatn < 4. Sincen > 3weconcludethatn = 3. Substituting n = 3andr = g−2
into equation (4.4)weobtain
3
2+ r
i
g > i=1 .
P3
−1+ 1
ri+1
i=1
Rearrangingwe obtain P
3 3
r +r +r +2
g > (r +1) 1 2 3 > (r +1).
i i
r +r +r +2−r r r
i=1 !(cid:18) 1 2 3 1 2 3(cid:19) i=1
Y Y
3
Since 1 < 1 we conclude, up to permutation of the r , that r = r = 1, r > 1 or
ri+1 i 1 2 3
i=1
r1 = 1,r2 P= 2 and 2 6 r3 6 4. Finally if r1 = r2 = 1 and r3 > 1 then the condition
3
(r +1) < g impliesthat 1 6 r 6 g −2. (cid:3)
i 3 4
i=1
Q
Havingproved Theorem 1.2, we nowdeduce Corollary 1.3 (b).
Proof of Corollary 1.3(b). Let C be a general curve of genus g. Suppose that C lies on the
image of some Segre embedding Pr1 × ··· × Prn → Pg−1. By Proposition 1.1, C admits
g n
line bundles L ,...,L such that L = K , h0(C,L ) = r + 1 and (r + 1) = g. By
1 n i C i i i
i=1 i=1
3 N Q
Theorem 1.2, n = 3 andg > (r +1). Thisisa contradiction. (cid:3)
i
i=1
Q
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8 NATHANGRIEVE
DEPARTMENT OF MATHEMATICS AND STATISTICS, QUEEN’S UNIVERSITY, KINGSTON, ON, K7L 3N6,
CANADA
E-mailaddress: [email protected]