Higher order embeddings of projective curves

Anna Stasica

IMUJ PREPRINT 2000/09

Abstract

In this note we show that every projective curve admits a k-very ample linear series of dimension at most 2k + 1. We also show how the order of the canonical embedding is related to the gonality of the given curve.

1 Introduction

Let X be a smooth projective variety of dimension n. Let L be a line bundle on X and let V ⊂ H^{0}(X, L) be a linear system of divisors. We say that V is k-very ample if the restriction map

V ⟶ H^{0}(X, L ⊗ O_{Z})

is surjective for every 0-dimensional subscheme Z ⊂ X of length k + 1. In particular L is k-very ample if V = H^{0}(X, L). Note that 0-very ampleness is equivalent to base point freeness and 1-very ampleness to the usual very ampleness. If V is k-very ample for k ≥ 0 then the associated mapping φ_{V} ⟶ ℙ(V^{∗}) is called an embedding of order k. Higher order embeddings play an important role in algebraic geometry, in particular in the investigation of local positivity of line bundles.

1991 Mathematics Subject Classification: 14C20, 14H60

Key words and phrases: canonical divisor, embedding, linear system, k-very ampleness

Our main goal is to generalise two basic results of the theory of projective curves and canonical embeddings. We prove the following theorems: Theorem 3.1. For any smooth projective curve there exists a k-very ample linear series V such that dim V ≤ 2k + 1.

Theorem 3.2. The canonical divisor K_{X} is k-very ample if and only if the curve X is not r-gonal for r = 1, …, k + 1. The first statement generalizes the well known fact that every projective curve can be embedded into ℙ^{3} (see e.g. [H]). The second one generalizes the result which says that the canonical divisor on a curve X of genus g ≥ 2 is very ample if and only if X is not hyperelliptic (Proposition 5.2, IV [H]).

1.1 Notation

In this paper we work over the field ℂ of complex numbers. We denote by K_{X} the canonical divisor of a smooth projective curve X. If L is a line bundle on X then by | L | we denote the complete linear series associated to L and by h^{0}(L) the dimension of the vector space H^{0}(X, L) of global sections of the bundle L. Similarly for an effective divisor D on X we denote the linear series | O_{X}(D) | by | D |. Linear equivalence of divisors D_{1}, D_{2} is denoted by D_{1} ∼ D_{2}

2 Auxiliary results

In this part we recall some useful properties of k-very ample divisor on curves. First we note that if X is a curve, V is k-very ample if an only if

V ⟶ H^{0}(X, L ⊗ (O_{X}/𝔪_{x_{1}}^{k_{1}} ⊗ ⋯ ⊗ 𝔪_{x_{r}}^{k_{r}}))

is surjective, where 1 ≤ r ≤ k + 1, k_{1} + ⋯ + k_{r} = k + 1 and 𝔪_{x_{i}} denotes the maximal ideal of the point x_{i} ∈ X for i = 1, …, r. In fact, a 0-dimensional subscheme of length k on a curve X consists of k points counted with multiplicity. In particular we have a nice description of k-very ampleness: the linear series | L | on curve X is k-very ample if and only if

dim | L − (x_{1} + ⋯ + x_{k+1}) |= dim | L | −(k + 1)  (1)

for any set of k + 1 points on X (possibly some of these points are equal). We say that k + 1 points impose independent conditions on divisors in the linear system | L |.

The following lemma gives a sufficient condition for a divisor on a smooth curve to be k-very ample.

Lemma 2.1 Let X be a smooth curve of genus g and let D be a divisor on X. If deg D ≥ 2g + k, then the linear series | D | is k-very ample.

Proof. Let P be a divisor of the form P = P_{1} + ⋯ + P_{k+1}, where P_{1}, … P_{k+1} are some points on X. By (1) it is enough to show that dim | D |= dim | D − P | +k + 1. Applying the Riemann-Roch theorem we get:

1. dim | D | − dim | K_{X} − D |= 1 − g + deg D

2. dim | D − P | − dim | K_{X} − D + P |= 1 − g + deg D − k − 1

but deg K_{X} − D = 2g − 2 − deg D < 0 so dim | K_{X} − D |= 0 and

dim | D |= 1 − g + deg D.

In the same way we can show that dim | K_{X} − D + P |= 0 hence | D | is k-very ample.

The next proposition shows that the inequality in previous lemma might not be sharp.

Proposition 2.2 Let L be a line bundle of degree 2g + k − 1 on a curve X of genus g. Then L is k-very ample if an only if h^{0}(L − K_{X}) = 0.

Proof. By the Riemann-Roch theorem we have an equality

h^{0}(L) = 1 − g + 2g + k − 1 = g + k.  (2)

Let P = P_{1} + … + P_{k+1}, where every P_{i} ∈ X is a point. Note that deg(K_{X} − L + P) = 0 and hence

h^{0}(L − P) = h^{0}(K_{X} − L + P) + g − 1.  (3)

If L is a k-very ample line bundle we get, due to (1) and (2), the equality

h^{0}(L − P) = h^{0}(L) − k − 1 = g − 1

and consequently h^{0}(K_{X} − L + P) = 0. It means that the linear system | L − K_{X} | is empty, otherwise we would have an effective divisor P′ such that L − K_{X} ∼ P′, hence h^{0}(K_{X} − L + P′) ≥ 1, but this fact would contradict the equality h^{0}(K_{X} − L + P) = 0. In consequence we get h^{0}(L − K_{X}) = 0. On the other hand if h^{0}(L − K_{X}) = 0 then the linear system | L − K_{X} | is empty and hence for every divisor P = P_{1} + … + P_{k+1} we have h^{0}(K_{X} − L + P) = 0. Combining the Riemann-Roch theorem with (2) and (1), we get the assertion.

Proposition 2.3 If L is a k-very ample line bundle on a curve X then:

1. h^{0}(L) ≥ k + 1

2. the equality holds if and only if X is isomorphic to ℙ^{1} and L ≃ O_{ℙ^{1}}(k).

Proof. If L is k-very ample then

h^{0}(L) = dim | L | +1 = dim | L − (k + 1)P | +k + 2 ≥ k + 1.

Assuming that L is a line bundle on X with h^{0}(L) = k + 1, we obtain that the image by the associated map φ_{L}(X) is a curve in ℙ^{k}. To show that X is rational it is enough to notice that deg φ_{L}(X) = k (Corollary 7.7, [M]). Let H ≃ ℙ^{k+1} ⊂ ℙ^{k} be a generic hyperplane. If H.X > k then H ∩ X is a 0 - dimensional subscheme of lenght H.X, which contains a subscheme Z′ of lenght k + 1. But then Z′ does not impose independent conditions on H^{0}(X, L). This implies that the image of φ_{L} is of degree k.

At the end we would like to recall a useful lemma, due to Beltrametti and Sommese (Lemma 2.2, [BS]).

Lemma 2.4 If L_{1} and L_{2} are k_{1}-, respectively k_{2}-very ample line bundles on a curve then | L_{1} ⊗ L_{2} | is (k_{1} + k_{2}) - very ample.

3 Main results

Now we are going to show that for any projective curve there exists an embedding of order k into ℙ^{2k+1}.

Let X ⊂ ℙ^{N} be a nondegenerate curve. Then generic k + 1 points on X span ℙ^{k} ⊂ ℙ^{N} for k ≤ N.

Consider S_{X}^{k} = {(P_{1}, …, ℙ_{k+1}, u) | u ∈< P_{1}, …, P_{k+1} >} ⊂ X^{k+1} × ℙ^{N} We will denote by pr_{2} projection of this set into space ℙ^{N} and call the closed set Sec^{k}(X) = \overline{pr_{2}(S_{X}^{k})} the k-secant variety of X. It is clear that dimension of this variety is not greater than 2k + 1.

Theorem 3.1 For any smooth projective curve there exists a k-very ample linear series V such that dim V ≤ 2k + 1.

Proof. Let X be a curve of genus g. Let D be any divisor of degree 2g + k on X. Due to lemma 2.1 the divisor D is k - very ample and hence the associated map φ_{D} is k-th order embedding into ℙ^{N} (N = dim | D |). If N ≤ 2k + 1 then we are done.

In the other case we consider the projection p_{x_{1}} from some point x_{1} ∈ ℙ^{N} \ Sec^{k}(X) (such a point exists because dim Sec^{k}(X) ≤ 2k + 1)

p_{x_{1}} : X ⟶ ℙ^{N−1}.

Note that any subseries V_{1} − x_{1} ⊂| D | of codimension 1, given by hyperplanes passing through the point x_{1} is k-very ample. Indeed, the equality

dim p_{x_{1}}(< P_{1}, …, P_{k+1} >) = dim < P_{1}, …, P_{k+1} >

holds thanks to the fact that x_{1} ∉ Sec^{k}(X).

We continue this process untill we get k-very ample linear series V = V_{s} with s = N − (2k + 1). This series gives an embedding of X of order k into the space ℙ^{2k+1} as we claimed.

Note that if we have a nondegenerate curve X ⊂ ℙ^{2k+1} then its secant variety is exactly ℙ^{2k+1}, therefore, there is no k-very ample linear series of dimension < 2k + 1 on such a curve. Put

l_{0} = min{l | Sec^{l}(X) = ℙ^{2k+1}}.

If l_{0} = k, then Sec^{k}(X) = ℙ^{2k+1}. If l_{0} > k, we can apply Example 1.6 in [Z] which says that

dim Sec^{l}(X) = min{2l + 1, 2k + 1}.

This gives Sec^{l}(X) = 2l + 1 for l < l_{0} and, particularly Sec^{k}(X) = 2k + 1.

Finally we will show that the order of canonical embedding of a curve is determined by its gonality and vice versa.

Let X be a curve of general type, i.e. g(X) ≥ 2. We have the following characterization.

Theorem 3.2 The canonical divisor K_{X} is k-very ample if and only if the curve X is not r-gonal for r = 1, …, k + 1.

Proof. Assume that K_{X} is k-very ample. Take P = x_{1} + … + x_{k+1}, where x_{1}, …, x_{k+1} ∈ X are any points. By Riemann-Roch theorem we have

dim | P |= 2 − g + k + dim | K_{X} − P |= 0

since dim | K_{X} − P |= g − 1 − (k + 1) by assumption. It means that any linear system of degree k + 1 has dimension 0 and hence X is not (k + 1) - gonal (and of course not r-gonal for r ≤ k). Now we will assume that the canonical divisor is not k-very ample. Then there exist points x_{1}, …, x_{k+1} ∈ X such that

dim | K_{X} − P |≥ dim | K_{X} | −k,

where P = x_{1} + … + x_{k+1}. Again by Riemann-Roch theorem, we have

dim | P |≥ 1 − g + k + 1 + g − 1 − k = 1.

Let y_{1}, …, y_{t} be all the base points of the linear system | P |. Than the system | P −y_{1} −…−y_{t} | is globally generated and its dimension is equal to the dimension of | P |. Since any generic subseries V ⊂| P − y_{1} − … − y_{t} | of dimension 1 is base points free we have a morphism φ_{V} : X → ℙ^{1} of degree at most k + 1 which implies that X is not (k + 1)-gonal.

References

[BFS] M.C. Beltrametti, P. Francia, A.J. Sommese. On Reider’s method and higher order embeddings. Duke Math. J., p. 425-439 (1989)

[BS] M.C. Beltrametti, A.J. Sommese. On k-Jet Ampleness. In: Complex Analysis and Geometry, edited by V. Ancona and A. Silva, Plenum Press, New York, 1993, pp. 355-375

[GH] P.Griffiths, J.Harris. Principles of Algebraic Geometry. Wiley Classic Library Edition Published 1994

[H] R. Hartshorne. Algebraic Geometry. Springer-Verlag 1977

[M] D. Mumford. Algebraic Geometry I Complex Projective Varieties. Izdatielstwo Mir, Moscow 1979 [Z] F.I. Zak. Tangents and secants of algebraic varieties. AMS, Provi-

dence, RI, 1993

Instytut Matematyki Uniwersytet Jagielloński ul. Reymonta 4, 30-059 Kraków E-mail: astasica@im.uj.edu.pl