Abstract
Abstract. For a finite dimensional vector space G we define the k-th generic syzygy scheme Gensyz k (G) by explicit equations. If X ⊂ P n is cut out by quadrics and f is a p-th syzygy of rank p + k + 1 we show that the syzygy scheme Syz(f ) of f is a cone over a linear section of Gensyz k (G). We also give a geometric description of Gensyz k (G) for k = 0, 1, 2, in particular Gensyz 2 (G) is the union of a Plücker embedded Grassmannian and a linear space. From this we deduce that every smooth, non-degenerate projective curve C ⊂ P n which is cut out by quadrics and has a p-th linear syzygy of rank p + 3 admits a rank 2 vector bundle E with det E = OC(1) and h 0 (E ) ≥ p + 4.
Introduction
Let X ⊂ P n be a projective variety that is cut out by quadrics. One can then look at the linear strand of its minimal free resolution and ask whether a p-th linear syzygy f carries some geometric information about X. For this purpose Ehbauer [Ehb94] introduced the syzygy scheme Syz(f ), which is cut out by the quadrics involved in f . The syzygy scheme always contains X and can be explicitly calculated in some cases. Ehbauer studied this construction when X is a set of points in uniform position.
Another geometric invariant of a p-th syzygy f is the space G * of linear forms involved in f . Its dimension is called the rank of f . Interesting syzygy varieties often arise from syzygies of low rank.
In [Sch91] Schreyer observed that for p = 1 the syzygy scheme Syz(f ) is always a cone over a linear section of a generic syzygy scheme Gensyz k with k = rank f − 2 and gave explicit equations for Gensyz k in this case. Eusen and Schreyer found a geometric description of these schemes for k ∈ {0, . . . , 4} and p = 1 in [ES94] .
In this paper we define more general generic syzygy schemes Gensyz k (G) by explicit equations depending on a finite dimensional vector space G. With these schemes we prove:
Theorem 3.4. Let I ⊂ R be a homogeneous ideal generated by quadrics and f a p-th rank p + k + 1 linear syzygy of I. Then the syzygy scheme Syz(f ) is isomorphic to a cone over a linear section of Gensyz k (G) where G is the space of (p − 1)-st syzygies involved in f . We also obtain a geometric description of Gensyz k (G) for k = 0, 1, 2 and arbitrary G. We show that Gensyz 0 (G) is always the union of a hypersurface with a point and that Gensyz 1 (G) is a Segre-embedded P 1 × P rank f −1 . The main new result of this paper is Theorem 6.1. Let G be a g dimensional vector space, then
where G(C ⊕ G * , 2) is the Grassmannian of two dimensional quotient spaces of C ⊕ G * . Moreover the second generic syzygy ideal I of G is reduced and saturated.
The geometric descriptions of Gensyz k (G) allow us to draw a number of conclusions:
Corollary 4.2. Let X ⊂ P n be a projective variety, I X generated by quadrics and f ∈ F p a p-th syzygy of rank p + 1. Then X is either contained in a hyperplane or reducible.
This result seems to be well known, but we include it since it follows directly from our methods.
Corollary 5.2. Let X ⊂ P n be a non-degenerate irreducible projective variety, I X generated by quadrics and f ∈ F p a p-th syzygy of rank p + 2. Then the syzygy scheme Syz(f ) of f is a scroll of degree p + 2 and codimension p + 1.
In particular a p-th syzygy of rank p + 1 implies the existence of a special pencil |D| on X cut out by the fibers of the scroll. If X is a canonical curve |D| has low Clifford index. These pencils are the ones that play a role in Green's conjecture [Gre84] . Our corollary above is therefore probably well known to experts in this field.
Our main new geometric result is
Theorem 6.7. Let C ⊂ P n be a smooth, irreducible non-degenerate curve. If C is cut out by quadrics and has a p-th syzygy f of rank p + 3, then there exists a rank 2 vector bundle E on C with det E = O C (1) and h 0 (E) ≥ p + 4.
In the case of a canonical curve these are rank 2 budles with canonical determinant.
One can also use the methods of this paper to construct the Mukai-Lazarsfeld bundle on a K3 surface directly from a syzygy f . This is the vector bundle that played a central role in Voisin's proof of Green's conjecture [Voi02] , [Voi03] . The Grassmannian used by Voisin in her proof is dual to the Grassmannian obtained as the generic syzygy scheme of f . This paper is structured as follows. In Section 2 we recall what we need about syzygies, syzygy ideals and syzygy schemes. In Section 3 we define the generic syzygy varieties and show that every syzygy scheme is a cone over a linear section of a generic syzygy scheme. In the last three sections we describe the k-th generic syzygy varieties for k = 0, 1, 2 geometrically and study syzygies of rank p + 1, p + 2 and p + 3.
Syzygies, Syzygy Ideals and Syzygy Schemes
For the purpose of this paper let R = C[x 0 , . . . , x n ] be the homogeneous coordinate ring of P n . With R(−i) we denote R with its grading shifted, i.e. R(−i) j = R j−i . Often we abbreviate the space of linear polynomials R 1 ⊂ R by V and write P n = P(V ) using the Grothendieck notation.
Definition 2.1. Let I ⊂ R be a homogeneous ideal, generated by quadrics, and
the linear part of the minimal free resolution of I. The elements of F i are called i-th linear syzygies of I.
Definition 2.2. Let I ⊂ R be a homogeneous ideal, generated by quadrics and f ∈ F p a p-th linear syzygy. We define the space of (p − 1)-st linear syzygies involved in f as the smallest vector space G ⊂ F p−1 such that there is a commutative diagram
We define the rank of f as the dimension of G.
The above diagram extends to a map from the Koszul complex of G to the linear strand of I:
Remark 2.3. Observe that by dualizing and twisting the morphism
from above, G * is exhibited as a space of linear forms on P n . We therefore call G * the space of linear forms involved in f .
Lemma 2.4. In the map of complexes of Definition 2.2 all vertical maps are nonzero.
Proof. Suppose there exists an integer k such that in the diagram
the morphism ϕ k−1 is zero, but ϕ k is nonzero. Then the image of ϕ k is a free summand of F k ⊗ R(−k − 2) which maps to zero in the linear strand of the minimal free resolution of I. This contradicts the minimality of the resolution.
Corollary 2.5. Let f be a p-th linear syzygy of I ⊂ R. Then rank f ≥ p+1.
Proof. If rank f ≤ p then p+1 G vanishes and the first vertical map of the map of complexes in Definition 2.2 would have to be zero. Definition 2.6. Let I ⊂ R be an ideal generated by quadrics, f ∈ F p a p-th linear syzygy and I f the syzygy ideal of f . Then the vanishing set Syz(f ) = V (I f ) is called the syzygy scheme associated to f .
Remark 2.7. Observe that Syz(f ) ⊂ P n is always a strict subset, since the syzygy ideal I f is never empty by Lemma 2.4.
Generic syzygy schemes
Definition 3.1. Let G be a vector space of dimension g and consider the ring
The ideal I defined by the natural inclusion
is called the k-th generic syzygy ideal of G. Its vanishing set Gensyz k (G) is called the k-th generic syzygy scheme of G. 
induces a commutative diagram of free S-modules
The top arrow is resolved by the minimal free resolution of I and the bottom arrow by the rest of the Koszul complex. Since both complexes are exact and minimal, the maps above lift to a map of complexes. This map is injective in each new step since it is injective in the F 0 step. For degree reasons, the image of this map of complexes must lie in the linear strand of I.
Corollary 3.3. The k-th generic syzygy scheme of G has a natural 1-dimensional space of rank g linear syzygies in step g − k − 1. The space of (g − k − 2)-nd syzygies involved in anyone of these is isomorphic to G.
Proof. The (g −k −1)-st syzygies given by Proposition 3.2 have rank at most g since g−1 G * ∼ = G has dimension g. The rank of these syzygies cannot be smaller, since the last map of the Koszul complex is surjective in degree g − k.
Theorem 3.4. Let I ⊂ R be a homogeneous ideal generated by quadrics and f a p-th rank p + k + 1 linear syzygy of I. Then the syzygy scheme Syz(f ) is isomorphic to a cone over a linear section of Gensyz k (G) where G is the space of (p − 1)-st syzygies involved in f .
Proof. We have the map of complexes
from Definition 2.2. Consider the map
given by mapping the elements of G * to their corresponding linear forms and the elements of k G * = p+1 G to their images under the map α. The induced diagram
and its degree 2 part
shows that ϕ maps the k-th generic syzygy ideal surjectively to the syzygy ideal I f of f . Projectively the image of ϕ defines a linear subspace
The calculation above shows that Syz(f ) is a cone over P(Im ϕ)∩Gensyz k (G) with vertex V (Im ϕ) ⊂ P(V ).
Reducible Syzygies
Proposition 4.1. Let G be a g dimensional vector space, then
i.e Gensyz 0 (G) is the union of a hyperplane and a point. Moreover the generic syzygy ideal of I of Gensyz 0 (G) is reduced and saturated.
Proof. The ideal of the hyperplane P(G * ) ∼ = P g−1 is generated by the linear forms in 0 G * ∼ = C. The ideal of the point P( 0 G * ) ∼ = P 0 is generated by the linear forms in G * . Since the two ideals involve different sets of variables, their intersection is the same as their product:
This is the 0-th generic syzygy ideal of G.
Corollary 4.2. Let X ⊂ P n be a projective variety, I X generated by quadrics and f ∈ F p a p-th syzygy of rank p + 1. Then X is either contained in a hyperplane or reducible.
Proof. By Theorem 3.4 and Proposition 4.1 Syz(f ) is a cone over a linear section of a hyperplane and a point. Since Syz(f ) can not contain all of P n by Remark 2.7, Syz(f ) ⊂ P n must be the union of a hyperplane and possibly a second linear subspace. Since X is contained in Syz(f ) it must be either reducible or contained in one of the two linear subspaces.
Definition 4.3. Let X ⊂ P n be a projective scheme, whose ideal is cut out by quadrics. A p-th linear syzygy of X is called reducible, if it has rank p + 1.
Scrollar Syzygies
Theorem 5.1. Let G be a g dimensional vector space, then
Moreover the second generic syzygy ideal I of G is reduced and saturated.
Proof. Observe that G * ⊗ (C ⊕ C) = G * ⊕ G * . We can therefore consider the Segre embedding
The ideal of P g−1 × P 1 is generated by the Segre quadrics:
This is the first generic syzygy ideal of G.
Corollary 5.2. Let X ⊂ P n be a non degenerate irreducible projective variety, I X generated by quadrics and f ∈ F p a p-th syzygy of rank p + 2. Then the syzygy scheme Syz(f ) of f is a scroll of degree p + 2 and codimension p + 1.
Proof. Let G be the g = p + 2 dimensional space of (p − 1)-st syzygies involved in f . By theorem 3.4 the syzygy scheme Syz(f ) is a linear section of a cone over P p+1 × P 1 . Since P p+1 × P 1 has codimension p + 1 and degree p + 2 in P(G * ⊕ Λ 1 G * ) we only have to prove that this intersection is of expected codimension. By Eisenbud [Eis95, Ex. A2.19] this is the case if the matrix M whose 2 × 2-minors cut out P p+1 × P 1 remains 1-generic after we apply the map ϕ :
from the proof of Theorem 3.4.
If ϕ(M ) is not 1-generic, we can choose bases of G * and C ⊕ C such that ϕ(M ) has the form
with l 1 , . . . , l p+1 a basis of G * and a 1 , . . . , a i linearly independent. Since the syzygy ideal I f cannot be empty by Lemma 2.4, i has to be at least 1. In this situation I f contains the 2 × 2 minor
which implies that X must be reducible or degenerate. This contradicts our assumptions.
Definition 5.3. Let X ⊂ P n be a projective scheme, whose ideal is cut out by quadrics. A p-th linear syzygy of X is called scrollar, if it has rank p + 2.
Example 5.4. Let C ⊂ P g−1 be a non hyperelliptic canonical curve of genus g and |D| a pencil of Clifford index cliff(D) = g − p − 3. The p-th syzygy of C constructed by the method of Green and Lazarsfeld in [GL84] is scrollar.
With the above geometric description of scrollar syzygy varieties one can prove the following well known converse of the Green-Lazarsfeld construction:
Proposition 5.5. Let C ⊂ P g−1 be a non hyperelliptic canonical curve of genus g and f ∈ F p a p-th scrollar syzygy. Then there exists a linear system
Proof. Let G * be the p + 2 dimensional space of linear forms involved in f . Then the syzygy scheme Syz(f ) of f is a scroll that contains C and has the vanishing set V (G * ) as a fiber. Set D = C ∩ V (G * ). Since C ⊂ P g−1 is non degenerate, D is a divisor on C. We consider the linear system |D|. Since D is cut out by the ruling of Syz(f ) we have h 0 (D) ≥ 2. Also h 0 (K −D) ≥ p+2 since the linear forms in G * cut out canonical divisors of C that contain D. Riemann-Roch now gives:
Remark 5.6. For general k-gonal canonical curves C Green's conjecture is equivalent to the claim that every step of the linear strand C contains at least one scrollar syzygy. This was recently shown by Voisin [Voi02] , [Voi03] .
More generally one can make the following conjecture This conjecture is known for for p = 1 when g = 8 and for p = 2 when g = 8 by [vB00] and [vB02] .
6. grassmannian syzygies Theorem 6.1. Let G be a g dimensional vector space, then
where G(C ⊕ G * , 2) is the Grassmannian of two dimensional quotient spaces of C ⊕ G * . Moreover the second generic syzygy ideal I of G is reduced and saturated.
Proof. Observe that 2 (C ⊕ G * ) = G * ⊕ 2 G * . We can therefore consider the Plücker embedding
and the ideal of the Grassmannian G which is generated by 4 × 4-pfaffians of a skew symmetric matrix. More precisely:
On the other hand P( 2 G * ) ∼ = P ( g 2 )−1 =: P is cut out by the linear forms in G * , so I P = (G * ). To prove the theorem we calculate the intersection of C has scrollar p-th syzygies for p ≤ g−4 2 . The rank of a syzygy f can fall by at most one when restricting to a general hyperplane (i.e. when the linear form defining H is involved in f ). Since X has no scrollar syzygies, the scrollar syzygies of C must come from grassmannian syzygies of X.
We now describe some geometric consequences of grassmannian syzygies. For this let Q be the universal rank 2 quotient bundle on the Grassmannian G = G(C⊕G * , 2). The global sections of Q are given by H 0 (G, Q) = C⊕G * .
Lemma 6.4. Let s ∈ H 0 (G, Q) be a global section and I s the ideal of its vanishing locus on G. Then I s is generated by hyperplane sections of G, more precisely
Proof. Consider the Koszul complex associated to s:
Since Q is globally generated, the converse also follows.
Remark 6.5. Observe that for a section s ∈ C ⊂ C ⊕ G * = H 0 (G, Q) we have I G + I P = I s . In other words a grassmannian syzygy f defines up to a constant a section of Q.
Lemma 6.6. Let X ⊂ P(V ) be a projective variety cut out by quadrics, f a p-th grassmannian syzygy of X, G the space of (p − 1)-st syzygies involved in f , and ϕ : G * ⊕ 2 G * → V the induced map. Then the natural map
is injective.
Proof. By construction Im ϕ contains G * so the non-zero elements of G * are not contained in I P(Im ϕ) . On the other hand the vanishing ideal
contains the whole space C ∧ G * = G * if s ∈ C, or a non-zero element of G * ∧ C = G * if s ∈ G * . So I s can never be contained in I P(Im ϕ) and H 0 (Q ⊗ I G∩P(Im ϕ)/G ) = 0. The proposition then follows from the exact sequence
Theorem 6.7. Let C ⊂ P n be a smooth, irreducible non-degenerate curve. If C is cut out by quadrics and has a p-th grassmannian syzygy f , then there exists a rank 2 vector bundle E on C with det E = O C (1) and h 0 (E) ≥ p + 4.
Proof. Let Syz(f ) be the syzygy scheme of f . By Theorems 3.4 and 6.1 Syz(f ) is a cone over a linear section of G(p + 4, 2) ∪ P ( p+3 2 )−1 . Now Syz(f ) contains C and C is irreducible and non-degenerate, so C must be contained in a cone Y over a linear section of G. The universal quotient bundle Q on G restricts to G ∩ P(Im φ) and pulls back to a rank 2 vector bundle
If C intersects the vertex of Y in a divisor, we consider the blowupỸ of Y in the vertex. Q then pulls back to a rank 2 vector bundle QỸ onỸ . Since C is smooth the strict transformC of C is isomorphic to C and QỸ restricts to a rank 2 vector bundle E onC ∼ = C.
Finally C can not be contained in the vertex of Y since C is non-degenerate.
By Lemma 6.6 we have h 0 (Q| G ∩ P(Im φ)) ≥ p + 4. These sections extend to Y • . By Lemma 6.4 the zero loci of sections of Q are cut out by linear forms and their closures contain the vertex of Y . Since X is non-degenerate it can not lie in one of these zero loci, so all sections of Q descend to sections of E.
Example 6.8. Our method can is some cases also be used to obtain vector bundles on varieties of higher dimension. Let for example X ⊂ P g be a K3 surface of even sectional genus g = 2k whose Picard group is generated by a general hyperplane section. Then X has a grassmannian (k − 2)-nd syzygy by the argument of Example 6.3. One can show that in this case the map
is surjective. Therfore Syz(f ) is not a cone, and Q restricts to a rank 2 vector bundle E on X with det E = O X (1) and h 0 (E) ≥ k + 2. This is the Mukai-Lazarsfeld bundle used by Voisin in her proof of Green's conjecture [Voi02] .
This example leads us to ask Question 6.9. Let X ⊂ P n be a surface cut out by quadrics whose Picard group is generated by a general hyperplane section. Does every step of the linear strand of X contain a grassmannian syzygy?
Remark 6.10. Voisin's Theorem about the syzygies of K3 surfaces in [Voi02] prove that the answer to this question is "yes" in the case of K3 surfaces X ⊂ P g with sectional genus g = 2k.
Even more generally we ask Question 6.11. Let X ⊂ P n be a surface cut out by quadrics whose Picard group is generated by a general hyperplane section. Is the space of p-th linear syzygies of X spanned by grassmannian syzygies?
Remark 6.12. The answer to this question is "yes" for general K3 surfaces X ⊂ P g with sectional genus g ≤ 8 by the methods of [vB02] 

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