Geometric Syzygies of Mukai Varieties and General Canonical Curves with Genus at most 8

We describe the spaces of minimal rank last syzygies for the Mukai Varieties of sectional genus 6,7 and 8. Based on this we show: 1. The first geometric syzygies of a general canonical curve of genus 6 form a non degenerate configuration of 5 lines in P^4. 2. The first geometric syzygies of a general canonical curve of genus 7 form a non degenerate, linearly normal, ruled surface of degree 84 on a spinor variety S in P^15. 3. The second geometric syzygies of a general canonical curve of genus 8 form a non degenerate configuration of 14 conics on a 2-uple embedded P^5 in P^20. This proves a natural generalization of Green's conjecture [1984], namely that the geometric syzygies should span the space of all syzygies, in these cases. We have generalized results 1 and 3 to general curves of even genus in math.AG/0108078. Result 2 is the main new result of this paper.

q q q q q q q q q q q q q q q q q q q q q q q q q q q q Green's Conj. =⇒ GSC ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ known by [AM67], [Gre84b], [vB00] ❜ unknown this article The starting point of our proof is Theorem (Mukai). Every general canonical curve of genus 7 ≤ g ≤ 9 is a general linear section of an embedded rational homogeneous variety M g . General canonical curves of genus 6 are cut out by a general quadric on a general linear section of a homogeneous variety M 6 .
Using this we first consider the schemes of minimal rank first syzygies (g = 6, 7) respectively minimal rank second syzygies (g = 8) of the Mukai varieties M g using representation theory. It turns out that all these schemes contain large rational homogeneous varieties.
Passing from Mukai varieties to canonical curves we describe their schemes of geometric syzygies as determinantal loci on the above homogeneous varieties.
Using the resolutions of Eagon-Northcott (for g = 6, 8) and Lascoux (for g = 7) we express the cohomology of the corresponding ideal sheafs in terms of the cohomology of homogeneous bundles. The later cohomology is then calculated with the theorem of Bott.
This calculation shows h 0 (I(1)) = 0, proving the geometric syzygy conjecture in these cases. More precisely our results are: 4.2.3 Theorem. The scheme Z of last scrollar syzygies of a general canonical curve C ⊂ P 5 of genus 6 is a configuration of 5 skew lines in P 4 that spans the whole P 4 of first syzygies of C.

Theorem.
The scheme Z of last scrollar syzygies of a general canonical curve C ⊂ P 6 of genus 7 is a linearly normal ruled surface of degree 84 on a spinor variety S + syz ⊂ P 15 . This ruled surface spans the whole P 15 of first syzygies of C.
6.2.3 Theorem. The scheme Z of last scrollar syzygies of a general canonical curve C ⊂ P 7 of genus 8 is a configuration of 14 skew conics that lie on a 2-uple embedded P 5 ֒→ P 20 . Z spans the whole P 20 of second syzygies of C.
The paper is organized as follows: In section 2 we cover some well known background material on syzygies. In particular we will introduce the rank of a syzygy, the scheme of minimal rank syzygies and the vector bundle of linear forms in subsection 2.1.
In subsection 2.2 we will show, that varieties with very low rank syzygies always lie on certain scrolls. The properties of these scrollar syzygies are studied and their connection with Brill-Noether-Theory is explained. Subsection 2.3 finally describes what happens to a minimal rank syzygy of a variety X when X is intersected with a general linear subspace. It will turn out that the rank of s can drop and that this rank can be calculated by a morphism of vector bundles α involving the vector bundle of linear forms.
Section 3 fixes the notations for the use of representation theory, and the remaining three sections treat curves of genus 6, 7 and 8 in turn. Starting from the respective Mukai varieties the proof of the geometric syzygy conjecture outlined above is given in full detail.
I would like to thank Kristian Ranestad for the many helpful discussions during my stay at Oslo University. It was there where most of the ideas of this work where born.
I dedicate this paper to my grandmother Lilly-Maria who introduced me to mathematics.

Background on Syzygies
In this section we will review some standard facts from the study of linear syzygies.

Syzygies of Low Rank
Let X ⊂ P(V ) be a irreducible non degenerate variety, I X generated by quadrics and the linear strand of its resolution 2.1.1. Definition. An element s ∈ V k is called a k-th (linear) syzygy of X. P(V * k ) is called the space of k-th syzygies.
Every linear syzygy s involves a well defined number of linearly independent linear forms. This number is called the rank of s. In a more formal way we have: 2.1.2. Definition. Let s ∈ V k be a syzygy and the map of vector spaces induced by ϕ k . Then the image of s under this mapφ is a map of vector spaces, and its image is called the space of linear forms involved in s. Furthermore rank s := rankφ(s) = dim s is called the rank of s. The zero set Z s of a syzygy is the linear space subspace of P(V ) where the linear forms involved in s vanish.
2.1.3. Example. Consider the rational normal curve C ⊂ P 3 with minimal resolution where dim V 0 = 3 and dim V 1 = 2. If s, t is a basis of V 1 and q 1 , q 2 , q 3 a basis of V 0 , the mapφ 1 is given bỹ With this, the linear forms involved in s are s = x, y, z , the rank of s is 3 and the Zero locus of s is just the single point Z s = (0 : 0 : 0 : 1).
To apply geometric methods to the study of low rank syzygies we projectivize the space of kth syzygies P(V * k ) and give a determinantal description of the space Y min of minimal rank syzygies. The linear forms involved in these syzygies defines a vector bundle on Y min : 2.1.4. Definition. On the space of kth syzygies P(V * k ) the map of vector spacesφ k induces a map of vector bundles that satisfies The determinantal loci Y r (ψ) ⊂ P(V * k ) of ψ are called schemes of rank r syzygies, since the syzygies in their support have rank ≤ r.
On the scheme of minimal rank syzygies Y min := (Y r min (ψ)) red the restricted map ψ| Y has constant rank r min . Therefore the image L := Im(ψ| Y min ) is a vector bundle. We call it the vector bundle of linear forms, since for all minimal rank syzygies s ∈ Y min . 2.1.5. Example. For the rational normal curve C ⊂ P 3 all first syzygies are of rank 3 so Y min = Y 3 = P(V * 1 ) = P 1 . The vector bundle of linear forms on this P 1 is the image of where ψ is given by the flipped syzygy matrix Since ψ is injective we have L = 3O P 1 (−1).
2.1.6. Definition. We say that a variety X as above satisfies the kth minimal rank conjecture, if the scheme of kth minimal rank syzygies Y min ⊂ P(V * k ) is non degenerate.
This conjecture is easy to verify for certain rational homogeneous varieties: 2.1.7. Proposition. Let X = G/P ⊂ P(V ) be a linearly normal homogeneous rational variety. If the induced representation is irreducible then the variety of minimal rank kth syzygies Y min of X contains the minimal orbit G/P k ⊂ P(V * k ) of this representation. In particular X satisfies the k-th minimal rank conjecture.
Proof. Since the embedding X ⊂ P(V ) is linearly normal, it induces a representation Since X is G invariant, this G also acts on the minimal free resolution of I X and therefore induces a representation Now the rank of a syzygy s ∈ P(V * k ) is invariant under coordinate transformations of V so that the space of minimal syzygies Y min ⊂ P(V * k ) is G invariant. It is also compact, so it has to contain a minimal orbit G/P k of G in P(V * k ). Since ρ k is irreducible,there is only one minimal orbit and this minimal orbit is non degenerate.
It will turn out later, that the Mukai varieties for g = 6, 7, 8 are of this type.

Scrollar Syzygies
In this section we want to establish a connection between syzygies of very low rank of a variety X, rational scrolls that contain X and pencils of divisors on X.
First we describe how to construct the equations of a rational scroll S containing X from a kth syzygy of rank k + 2. Conversely we find that the minimal rank kth syzygies of this scroll are of rank k + 2. In fact there is a 1 : 1 correspondence between the fibers of S and its minimal rank kth syzygies.
Secondly we observe that we can construct a scroll S containing X from a pencil of divisors on X. Conversely the fibers of S will cut out a pencil of divisors on X.
It will turn out, that in the case minimal degree complete pencils of a general canonical curve C ⊂ P g−1 all these correspondences are the same. We will construct an isomorphism from the variety of minimal rank syzygies to the corresponding Brill-Noether locus in this case.
Except for the construction of the above morphism all results of this section are well known.
Lets start with 2.2.1. Proposition. Let X ⊂ P(V ) be a variety as above and s ∈ V k a kth syzygy of rank k + 2. Then there exists a rational scroll S ⊂ P(V ) of degree k + 2 and codimension k + 1 that contains X. Furthermore the vanishing set Z s ⊂ P(V ) is a fiber of S.
Proof. Let {x 1 , . . . , x k−2 } be a basis of s ⊂ V . Consider the Koszul complex where the maps are induced by multiplication with the trace element where α is again given by the multiplication with x * i ⊗ x i and β can be described by the multiplication with x * i ⊗ y i where {y 1 , . . . , v k+2 } ∈ V are linear forms. We now claim that the image of α • β is given by the 2 × 2-minors of More precisely we claim Without restriction we can assume (a, b) = (1, 2) and check So the 2 × 2 minors of M are contained in I X . Therefore the variety S cut out by these minors contains X.
Furthermore M is 1-generic. Suppose not, then M has after row and columnoperations the form M = x 1 x 2 . . . 0 y 2 . . . and X ⊂ Z(x 1 y 2 ) ⇐⇒ X ⊂ Z(x 1 ) or X ⊂ Z(y 2 ) since X is irreducible. This is impossible, since X is non degenerate.
Consequently M is 1-generic and S is a scroll as claimed above.
Conversely we have 2.2.2. Lemma. Let S ⊂ P(V ) be a scroll of degree k + 2 and codimension k + 1, k ≥ 1. Then the minimal rank kth syzygies of S have rank k + 2 and the space of minimal rank kth syzygies is isomorphic to the k-uple embedding of P 1 . Furthermore there is a 1 : 1 correspondence between minimal rank syzygies and fibers of S.
Proof. S is cut out by the 2 × 2-minors of a 1-generic matrix be the corresponding map of vector spaces, where dim F = k+2 and dim G = 2. Then I S is resolved by the Eagon-Northcott complex On the space of kth syzygies P(V * k ) ∼ = P(Λ k+2 F ⊗ Λ 2 G ⊗ S k G) * ∼ = P k the group GL(2) acts by coordinate transformation of G. The rank of a syzygy is invariant under such operations and therefore the space of minimal rank syzygies Y min ⊂ P k is invariant under GL(2). Furthermore Y min is compact since we are considering syzygies of minimal rank. Consequently every component of Y min must contain the minimal orbit To show Y = P 1 we calculate the tangent space of Y min in We recall the determinantal description of Y min . Consider the map induced by ϕ kφ Since Φ(f i , g j ) ∈ V this induces a map of vector bundles Consider the basis of V * k−1 which is dual to In this basis ψ is then given by the matrix In these coordinates we have s = (1 : 0 : · · · : 0) and therefore s = Im(Ψ| s ) = x 1 , . . . , x k+2 .
Since the matrix A was 1-generic, the above linear forms are linearly independent. This shows that rank s = k + 2 for all s ∈ P 1 ⊂ P k . Now consider the tangent vectors s ǫ = (1 : ǫ 1 : · · · : ǫ k ) Since we can without restriction suppose, that y 1 is linearly independent from x 1 , . . . , x k+2 , and since the syzygies of Y all have rank k + 2, all (k + 3) × (k + 3)-minors of the first k + 3 columns of the above matrix have to vanish for s ǫ : must vanish, proving ǫ i = 0 for i ≥ 2. Therefore the tangent space of Y min in s can be at most 1-dimensional. By applying GL(2) we get the same result for all points of P 1 ⊂ Y min proving P 1 = Y min .
The 1 : 1 correspondence is seen as follows.
The fibers of S are the vanishing sets of the generalized rows The above propositions suggest the following definition: 2.2.3. Definition. Let X be an irreducible, non degenerate variety and I X generated by quadrics. Then the kth syzygies of rank k+2 are called scrollar syzygies. The schemes Y k+2 of these syzygies are called spaces of scrollar syzygies.
If Y k+2 is nonempty, then this is also the space of minimal rank syzygies Y min since an irreducible, non degenerate variety can not have kth syzygies of rank k + 1.
2.2.4. Remark. Scrollar syzygies are the easiest example of the geometric syzygies constructed by Green and Lazarsfeld in [GL84].
We can now make a precise statement of the geometric syzygy conjecture for general canonical curves.
2.2.5. Conjecture (Geometric Syzygy Conjecture). Let C ⊂ P g−1 be a general canonical curve of genus g. Then all minimal rank syzygies are scrollar, and the spaces of scrollar syzygies are non degenerate.
2.2.6. Remark. For special canonical curves it is important to consider the non reduced scheme structure on the space of scrollar syzygies as can be seen in the case of a curve of genus 6 with only one g 1 5 [AH81, p. 174].
Also there are geometric kth-syzygies in the sense of Green and Lazarsfeld [GL84] which are not of rank k + 2. These must also be considered in the case of special curves. The easiest example of this phenomenon is exhibited by the plane quintic curve of genus 6 [vB00].
We now turn to the connection between scrolls and pencils. Here we restrict ourselves to the case of a canonical curve C ⊂ P g−1 . Let |D| be a complete pencil of degree d on C. Then we can consider the union 2.2.7. Proposition. S |D| is a rational normal scroll of codimension g − d containing C.
Proof. Since C is canonically embedded, Its equations are given by the 2 × 2-minors of the 2 × (g − d + 1)-matrix obtained from the natural map Conversely consider a scroll containing C. Its fibers cut out a pencil of divisors on C. These pencils are not always complete: 2.2.8. Proposition. Let C ⊂ P g−1 be a non hyperelliptic canonical curve of genus g contained in a scroll S of codimension c. Let F be a fiber of S and D = C.F . Then |D| is a g r d with r ≥ 1 and d ≤ g + r − c − 1.
Proof. The fibers of S cut out a pencil of divisors linearly equivalent to D. Therefore r = dim |D| ≥ 1.

The codimension of a fiber
In particular these linear systems have low Clifford index: 2.2.9. Corollary. In the situation above we have cliff(D) ≤ g − c − 2. If the corresponding complete linear system |D| is not a pencil, we even have Proof.
Proof. Let s ∈ V k be a scrollar syzygy in step k. Then the corresponding scroll S s has codimension k + 1 by proposition 2.2.1. The divisor D cut out by a fiber of S has Clifford index On the other hand it is well known, that on a general canonical curve all divisors have Clifford index at least ⌈ g−2 2 ⌉. Therefore 2.2.11. Corollary. The scrollar syzygy conjecture implies Green's conjecture for general canonical curves.
Proof. Assume the scrollar syzygy conjecture. Then all minimal rank syzygies are scrollar. But by the corollary above there are no scrollar syzygies in step k > ⌈ g−5 2 ⌉. Therefore there can be no syzygies at all in theses steps. This is Greens conjecture for the general canonical curve.
We want now to consider the last step in the resolution of a general canonical curve, that still allows syzygies: 2.2.12. Definition. Let C ⊂ P g−1 be a general canonical curve. Then the scrollar ⌈ g−5 2 ⌉th syzygies of C are called the last scrollar syzygies of C.
For the last scrollar syzygies everything is as nice as possible. First we calculate the degree of the corresponding divisors: 2.2.13. Lemma. Let C ⊂ P g−1 be a general canonical curve, a last scrollar syzygy, S the corresponding scroll and D the divisor cut out by the fiber F s corresponding to s. Then |D| is a complete pencil of degree ⌈ g+2 2 ⌉.
Proof. Suppose |D| was not a complete pencil. Then by corollary 2.2.9 we would have which is impossible for a general canonical curve. Consequently we have r = 1 and cliff D = ⌈ g−2 2 ⌉ the minimum possible value. In particular This allows us to construct a morphism from the space of last scrollar syzygies to the corresponding Brill-Noether-Locus: 2.2.14. Proposition. Let C ⊂ P g−1 be a general canonical curve, and Y min its scheme of last scrollar syzygies. Then there exists an isomorphism In particular Y min is a disjoint union of 2 g + 2 g g 2 rational curves if g is odd, and an irreducible ruled surface over W 1 Proof. Consider the vector bundle of linear forms L on the variety of Y min of last scrollar syzygies. Let Q be the cokernel of the natural inclusion Q is globally generated and has rank ⌈ g 2 ⌉. It therefore induces a morphism is the Grassmannian of ⌈ g 2 ⌉ dimensional quotient spaces of V , or equivalently the Grassmannian of ⌈ g−2 2 ⌉ dimensional linear subspaces of P g−1 .
Now consider the incidence Variety D is a family of divisors. The fiber over a scrollar syzygy s is the divisor cut out by the zero locus Z s of s. Lemma 2.2.13 shows that these divisors all have degree d = ⌈ g+2 2 ⌉ and r = 1. By the universal property of C r d we obtain a morphism To prove the surjectivity of ζ take let D ∈ C 1 ⌈ g+2 2 ⌉ be a divisor. The scroll S |D| spanned by |D| has codimension g−⌈ g+2 2 ⌉ and the fiber D corresponds to a scrollar ⌈ g−5 2 ⌉th (last) syzygy s with Z s = D by lemma 2.2.2. This implies D ⊂ Z s .C. Equality follows since they have the same degree by lemma 2.2.13.
We are left to prove that ζ is injective. Assume s, t are two last scrollar syzygies, whose zero sets Z s and Z t cut out the same divisor D = Z s .C = Z t .C. Then the scroll S |D| obtained from the complete pencil |D| is contained in the scrolls S s and S t corresponding to s and t. Now all these scrolls are of the same dimension, so they have to be equal. Since there is a 1 : 1 correspondence between divisors D ′ ∈ |D|, fibers s D ′ and scrollar syzygies of S |D| = S s = S t we must have s = t.
So ζ is bijective. Now Y min is reduced by definition, and since C is a general canonical curve, Now the Abel-Jacobi map 2 ⌉ is a disjoint union of finitely many P 1 's for g even and a ruled surface over W 1 ⌈ g+2 2 ⌉ for g odd. In the even case the number of P 1 's can be calculated by a formula of Castelnuovo [ACGH85, p. 211]:

Linear Sections
In this section we want consider general linear sections X ∩ P(W ) of X. It is well known that such a general linear section has syzygy spaces of the same dimension as X. So one can consider a syzygy s of X also as a syzygy of X ∩ P(W ). The rank of this syzygy can change however, if P(W ) does not intersect the zero locus Z s of s in the expected codimension.
This will lead to a determinantal description of those minimal rank syzygies of X that drop rank further when considered as syzygies of X ∩ P(W ).
If X is a Mukai-Variety and C = X ∩ P g−1 a general canonical curve it will turn out in the remaining sections of this paper, that these determinantal subvarieties are of expected dimension and describe the full space of last scrollar syzygies of C.
Let now X ⊂ P(V ) be an irreducible, non degenerate variety, I X generated by quadrics, P(W ) ⊂ P(V ) a linear subspace and π : V → W the corresponding projection. Consider the intersection X ∩ P(W ) and the linear strand of its resolution.
We start by recalling: If X is arithmetically Cohen Macaulay and X ∩ P(W ) is of the expected dimension, then all π k 's are isomorphisms.
Proof. Since X is arithmetically Cohen Macaulay, we have in particular H 1 (X, O X (q)) = 0 for all q ≥ 0. We can then apply [Gre84a, Thm 3.b.7] to get the result.
If we regard a syzygy s ∈ Y min (X) as a syzygy of X ∩ P(W ) in the way made precise by the preceding proposition, we can calculate the rank of s there by using the vector bundle of linear forms: 2.3.2. Corollary. Let X be ACM, X ∩ P(W ) of expected dimension, Y min (X) ⊂ P(V * k ) ∼ = P(W * k ) the scheme of k-th minimal rank syzygies of X and L the vector bundle of linear forms on Y min (X). Then there exists a map of vector bundles such that the rank of a syzygy s ∈ Y min (X) considered as a syzygy of X ∩ P(W ) is Since π k and π k−1 are isomorphisms, this induces a diagram on Y min the top map factors over L yielding Since β is surjective, we have for every syzygy s ∈ Y min rank(ψ W | s ) = rank(α| s ) as claimed.
A stronger statement is true for a general intersection with quadric hypersurface. Here the dimension of syzygy spaces and the rank of the syzygies stay the same: 2.3.3. Proposition. Let X ∈ P(V ) be an irreducible variety, I X generated by quadrics and X ∩ Q the intersection with a general quadric Q ⊂ P(V ). If is the linear strand of the resolution of I X , then is the linear strand of the resolution of I X∩Q . All differentials are the same except for ϕ 0 whose matrix has one more column of zeros.
In particular the kth syzygies (k ≥ 1) of X ∩ Q are the same as those of X and they have the same ranks.
Proof. We prove the proposition on the ring level. Let R = C[V ] be the coordinate ring of P(V ) and the graded minimal free resolution of R/I X . R/QR is resolved by the Koszul complex For degree reasons the linear strand of C • (Q) is the same as the one of C • except for the first step.
Later we will apply the last two propositions to Mukai varieties, as defined by 2.3.4. Theorem (Mukai). Every general canonical curve of genus 7 ≤ g ≤ 9 is a general linear section of an embedded rational homogeneous (Mukai) variety M g . General canonical curves of genus 6 are cut out by a general quadric on a general linear section of a homogeneous (Mukai) variety M 6 .
More explicitly we have the Grassmannian Gr(2, 5) ⊂ P 9 7 the Spinor-Variety S 10 ⊂ P 15 8 the Grassmannian Gr(2, 6) ⊂ P 14 9 the symplectic Grassmannian Gr(3, 6, η) ⊂ P 13 Proof. [Muk92b] [Muk92a] 3 Representation Theory As the Mukai varieties are rational homogeneous, the main tool of our study will we representation theory. We will use the following notations from Fulton/Harris [FH91] and Ottaviani [Ott95]: 3.1. Notation. We will denote by G a semisimple and simply connected Lie group P ⊂ G a parabolic subgroup p ⊂ g the corresponding Lie algebras h ⊂ p ⊂ g a Cartan subalgebra a decomposition into positive and negative roots the Cartan decomposition ∆ = {α 1 , . . . , α k } ⊂ R + the set of simple roots ω 1 , . . . , ω k the corresponding fundamental weights Σ ⊂ ∆ a subset of simple roots the corresponding parabolic subgroup of G S P (Σ) the semisimple part of p(Σ) W the Weyl group of g We also need the notion of a highest weight vector: is called a highest weight vector. λ ∈ h * is then called a highest weight.
With this we use further notations 3.3. Notation. We denote by ρ λ the irreducible representation of g with highest weight λ S λ the Schur functor for a partition λ Λ λ the Schur functor for the dual partitionλ 3.4. Remark. With this notation we have for example where Λ 3 is the usual exterior product.
If ρ is a representation of parabolic subgroup P ⊂ G with highest weight λ = λ 1 L 1 + · · · + λ n L n we use the notation E ρ , E(λ) or E(λ 1 , . . . , λ n ) for the vector bundle induced by ρ on G/P .

Theorem (Matsushima).
A vector bundle E of rank r over G/P is homogeneous if and only if there exists a representation ρ : Proof. [Ott95, Theorem 9.7] 3.6. Theorem (Classification of irreducible bundles over G/P ). Let P (Σ) ∈ G be a parabolic subgroup and ω 1 , . . . , ω k the fundamental weights corresponding to the subset of simple roots Σ ⊂ ∆. Then all irreducible representations of P (Σ) are where V is a representation of S P and n i ∈ Z. L ω i are the one dimensional representations of S P induced by the fundamental weights.
The weight lattice of S P is embedded in the weight lattice of G. If λ is the highest weight of V , we will call λ + n i w i the highest weight of the irreducible representation of P (Σ) above. In several proofs of this paper we need to calculate the decomposition of tensor products of fundamental representations. Formulas for this can be found in [KN88]. Also we sometimes calculate the dimension of certain representations. This can be done in various combinatoric ways and by the use of the Weyl character formula. In this paper all calculations of this kind have been checked by the computer program SYMMETRICA [KKL92] which can be used online on the web.

Genus 6
The following is well known, but sets the stage for the more involved computations for Mukai varieties of higher genus.

Syzygies of M 6
Let V be a 5-dimensional vector space with basis {v 1 , . . . , v 5 }. In this section we will abbreviate Λ λ V by Λ λ . Proof. The Grassmannian Gr(5, 2) is cut out by the 4 × 4 Pfaffians of a generic 5 × 5 matrix A. It is therefore a codimension 3 Gorenstein variety and has the resolution

The Mukai variety for genus 6 is
Since dim V = 5 this gives the above syzygy numbers.
Since GL(5) acts on Gr(5, 2) we also have an action on the syzygies. The Proof. From above we have a linear strand where the irreducible components have dimensions 5, 45 and 50 respectively.
where the irreducible components have dimensions 5 and 45 respectively. This implies V 1 = Λ 51 .
This allows us to describe the minimal rank syzygies Y min of Gr(5, 2) and the vector bundle of linear forms on Y min 4.1.3. Proposition. The scheme of minimal rank first syzygies of Gr(5, 2) is The bundle of linear forms on Y min is L| Y min = E(1, 1, 0, 0, 0) * = T P 4 (−2) L has rank 4.
Here this orbit GL(5)/P is the whole P 4 such that Y min = P 4 .
To describe the vector bundle of linear forms on Y min = GL(5)/P we have to determine the action of P on a fiber of L. We start by considering the dual actions ρ * of GL(5) and P on Λ 51 . The parabolic subgroup P in its standard representation is then the set of matrices that fix a given P 0 , i.e matrices of the form       * * * * * 0 * * * * 0 * * * * 0 * * * * 0 * * * *       The semisimple part of P is S P = GL(1) × GL(4) where GL(1) acts on v 1 and GL(4) acts on v 2 , v 3 , v 4 , v 5 in the standard way.
The maximal weight vector is a syzygy in Y min . To determine the fiber L| s of L over s we restrict the map from definition 2.1.4 to s. This gives Using Young-diagrams we get Consequently the fiber L| s of the line bundle of linear forms is and L is of rank 4. S P acts irreducibly on this fiber, and v 1 ∧ v 2 is the maximal weight vector of weight L 1 + L 2 . (Notice that the weights with respect to G are the same as the ones with respect to S P since we can use the same Cartan subalgebra h ⊂ LieS P ⊂ p ⊂ g).

General Canonical Curves of Genus 6
Let C be a general canonical curve of genus 6. From Mukai's theorem we obtain a P 5 ∼ = P(W ) ⊂ P 9 ∼ = P(Λ 2 V ) and a quadric in P 5 such that is a Del Pezzo surface and C = S ∩ Q 4.2.1. Proposition. On P 4 ∼ = Y min (Gr(5, 2)) there exists a map of vector bundles α : T P 4 (−2) → 6O P 4 such that its rank 3 locus Z 3 (α) is the scheme Z of last scrollar syzygies of C. Z is a configuration of 5 skew lines in P 4 .
Proof. ⌈ g−5 2 ⌉ = 1 so the first scrollar syzygies of C are also the last scrollar syzygies. The minimal rank first syzygies of Gr(5, 2) are of rank 4 and fill the whole space of first syzygies P(Λ * 51 ) = P 4 as calculated in proposition 4.1.3.
Since S is a general linear section of Gr(5, 2) we can apply corollary 2.3.2 to obtain a map α : L → W ⊗ O Y min whose rank calculates the rank of syzygies s ∈ Y min considered as syzygies of S. In our case this is equal to α : T P 4 (−2) → 6O P 4 since P(W ) = P 6 and L = T P 4 (−2). Now C is S intersected with a general quadric and therefore α also calculates the rank of s considered a a syzygies of C by proposition 2.3.3.
The last scrollar syzygies of C are first syzygies of rank 3. The argument above shows that the scheme Z of last scrollar syzygies contains the rank 3 locus Z 3 (α) of α.
Since P(W ) ⊂ P(Λ 2 V ) is a general subspace, and L * = Ω P 4 (2) is globally generated, Z 3 (α) is reduced and of expected dimension On the other hand we are also in the situation of corollary 2.2.14 which gives an isomorphism This shows that Z 3 (α) ⊂ Z is the union of at most 5 disjoint lines.
Since Z 3 (α) is reduced this shows that Z 3 (α) contains all 5 lines of Z. In particular we have Z=Z 3 (α).

Corollary. The ideal sheaf I Z/P 4 is resolved by
Proof. Since Z = Z 3 (α) and α drops rank in the expected dimension, the ideal sheaf is resolved by the corresponding Eagon-Northcott complex Since dim W * = 6 we have This gives the above multiplicities. Applying these equations to the complex above yields the desired resolution.

Theorem.
The scheme Z of last scrollar syzygies of a general canonical curve C ⊂ P 5 of genus 6 is a configuration of 5 skew lines in P 4 that spans the whole P 4 of first syzygies of C.

Remark.
Notice that h 1 (I Z/P 4 (1)) does not vanish. Therefore Z is not linearly normal. where G = Spin(V ) = Spin(10) is the 2 : 1 spin covering of SO(10), g = Lie G = so 10 its Lie algebra and spin + 5 a irreducible 16-dimensional spinor representation of G.

Proposition.
The linear strand of the resolution of S + 10 is Proof. From above we have the linear strand with dim V 0 = 10 and dim V 1 = 16. Now V 0 is a Spin(10) invariant subset of quadrics in P(spin + 5 ): where Λ + 5 is the irreducible representation corresponding to the maximal weight vector L 1 + · · · + L 5 . The representations have dimension 126,120 and 10 respectively. Therefore V 0 = Λ 1 . (For the decomposition of the tensor products see [KN88]).
In the next step we know where λ 1 · spin + 5 denotes the irreducible representation obtained by adding the maximal weights of Λ 1 and spin + 5 . The irreducible summands have dimensions 144 and 16 so that V 1 must be equal to spin − 5 .
5.1.3. Proposition. The scheme Y min ⊂ P(spin + 5 ) = P 15 of minimal rank first syzygies of the Spinor variety S + 10 contains an isomorphic Spinor variety The bundle of linear forms on S + syz is where B(−1) is the tautological quotient bundle on S + syz . B(−1) is of rank 5.
This is the spinor variety S + syz ⊂ P 15 .
To describe the vector bundle of linear forms L on S + syz we have to determine the action of P on a fiber of L. We start by considering the dual actions ρ * of Spin(10), P and S π(P ) on spin − 5 . The lie group of S π(P ) is Let P(W ) ⊂ Q be the P 4 left invariant by P With this we have the natural representations where Lie S P acts in the natural way. The w I = ∧ i I w i are weight vectors of G and P and their weights are 1 2 ( i∈I L i − i ∈I L i ). [FH91][pp. 305-306]. The maximal weight vector Consequently the fiber of the line bundle of linear forms is and L is of rank 5. S P acts irreducibly on this fiber, and w 1 ∧ w 2 ∧ w 3 ∧ w 4 is the maximal weight vector with weight 1 2 (L 1 + L 2 + L 3 + L 4 − L 5 ). Consequently

General Canonical Curves of Genus 7
Consider a general canonical curve C of genus 7. Mukai's Theorem provides us with P 6 = P(W ) ⊂ P(spin + 5 ) ∼ = P 15 (a different W from the last section) such that C = S + 10 ∩ P 6 5.2.1. Proposition. On the spinor variety S + syz ⊂ Y min (S + 10 ) there exists a map of vector bundles α : B(−1) → 7O S + syz such that its rank 3 locus Z 3 (α) is the scheme Z of last scrollar syzygies of C. Z is a ruled surface of degree 84.
Proof. ⌈ g−5 2 ⌉ = 1 so the first scrollar syzygies of C are also the last scrollar syzygies. The minimal rank first syzygies s ∈ S + syz of S + 10 are of rank 5 as calculated in proposition 5.1.3.
Since C is a general linear section of S + 10 we can apply corollary 2.3.2 to obtain a map α : L → W ⊗ O S + syz whose rank calculates the rank of syzygies s ∈ S + syz considered as syzygies of C. In our case this is equal to The last scrollar syzygies of C are first syzygies of rank 3. The argument above shows that the scheme Z of last scrollar syzygies contains the rank 3 locus Z 3 (α) of α.
Since P(W ) ⊂ P(spin + 5 ) is a general subspace, and L * = Ω B * (1) is globally generated, Z 3 (α) is reduced and of expected dimension On the other hand we are also in the situation of corollary 2.2.14 which gives an isomorphism ζ : Z → C 1 5 with C 1 5 a ruled surface over W 1 5 . Since C 1 5 ∼ = Z is irreducible, this shows that Z 3 (α) ⊂ Z is in fact an equality. In particular Z is a ruled surface.
Since Z = Z 3 (α) is of expected dimension and we can calculate its class with Porteous formula [ACGH85][p.86]: The cohomology ring of S + syz has been determined by Ranestad where h is the class of a hyperplane section and b is the third Chern class of B * . They also give the Chern polynomial of B * as The Chern polynomials needed for Porteous formula above are as obtained by the tautological sequence and c t (7O P 4 (1)) = (1 + Ht) 7 .
Proof. Since α : L → 7O Y drops rank in the expected dimension dim Z 3 (α) = 10 − (5 − 3)(7 − 3) = 2 the resolution of I Z/S + syz can be calculated using the methods of Lascoux. In the notation of [Las78, Thm 3.3] the above resolution is k(α, 2, 0) since α drops rank by two on Z 3 (α). This gives 5.2.4. Theorem. The scheme Z of last scrollar syzygies of a general canonical curve C ⊂ P 6 of genus 7 is a linearly normal ruled surface of degree 84 on a spinor variety S + syz ⊂ P 15 . This ruled surface spans the whole P 15 of first syzygies of C.
For the first bundle we have no vanishing root, and all positive roots α satisfy (α, λ + δ) > 0. Therefore i 0 = 0 and all higher cohomology vanishes for O(1). For the last bundle there are 9 positive roots α with (α, λ+ δ) < 0. Consequently the only non vanishing cohomology is H 9 (S 44222 L(1)). For all remaining bundles there is at least one integer that appears with opposite signs. Therefore we have a root α = L i + L j with (α, λ + δ) = 0, and all cohomology groups of theses bundles vanish.
Chasing the diagram (1)) = h 1 (I Z/S + syz (1)) = 0. This shows that the space of scrollar syzygies of C is non degenerate and linearly normal.

Genus 8
This case is very similar to the genus 6 case, since the Mukai variety is also a Grassmannian Gr(V, 2) but with dim V = 6. Let {v 1 , . . . , v 6 } be a basis of V .

Syzygies of M 8
The Mukai variety for genus 8 is with dim V 0 = 15, dim V 1 = 35 and dim V 2 = 21. V 0 is a invariant (not necessarily irreducible) subspace of quadrics. This gives where the irreducible components have dimensions 15, 105 and 105 respectively. So V 0 = Λ 4 . Similarly we have where the irreducible components have dimensions 1, 35 and 175 respectively. This implies V 1 = Λ 51 .
Finally we observe where the irreducible components have dimensions 21, 15, 384 and 105 respectively. Consequently we have V 2 = Λ 611 .
Using this we get 6.1.3. Proposition. The scheme of minimal rank second syzygies of Gr(6, 2) contains the minimal orbit The bundle of linear forms on P 5 is L has rank 5.
To describe the vector bundle of linear forms on P 5 = GL(6)/P we have to determine the action of P on a fiber of L. We start by considering the dual actions ρ * of GL(6) and P on Λ 611 . The parabolic subgroup in its standard representation is then the set of matrices that fix a given P 0 , i.e matrices of the form         * * * * * * 0 * * * * * 0 * * * * * 0 * * * * * 0 * * * * * 0 * * * * * The semisimple part of P is S P = GL(1) × GL(5) where GL(1) acts on v 1 and GL(5) acts on v 2 , v 3 , v 4 , v 5 , v 6 in the standard way.
The maximal weight vector

General Canonical Curves of Genus 8
Let now C be a general canonical curve of genus 8. From Mukai's Theorem we obtain a P 7 ∼ = P(W ) ⊂ P(Λ 2 ) ∼ = P 14 such that C = Gr(6, 2) ∩ P 7 6.2.1. Proposition. On P 5 ֒→ P 20 there exists a map of vector bundles α : T P 5 (−2) → 8O P 5 such that its rank 4 locus Z 4 (α) is the scheme Z of last scrollar syzygies of C. Z is a configuration of 14 skew conics on the 2-uple embedding P 5 ֒→ P 20 .
Proof. ⌈ g−5 2 ⌉ = 2 so the second scrollar syzygies of C are the last scrollar syzygies. The minimal rank second syzygies of Gr(6, 2) are of rank 5 and fill at least a 2-uple embedded P 5 ֒→ P 20 as calculated in proposition 6.1.3.
Since C is a general linear section of Gr(6, 2) we can apply corollary 2.3.2 to obtain a map α : L → W ⊗ O Y min whose rank calculates the rank of syzygies s ∈ Y min considered as syzygies of C. In our case this restricts to α : T P 5 (−2) → 8O P 5 on our P 5 .
The last scrollar syzygies of C are second syzygies of rank 4. The argument above shows that the scheme Z of last scrollar syzygies contains the rank 4 locus Z 4 (α) of α.
On the other hand we are also in the situation of corollary 2.2.14 which gives an isomorphism This shows that Z 4 (α) is the union of at most 14 disjoint P 1 's. Each of these P 1 is the scheme of second minimal rank syzygies of a scroll. These schemes are rational normal curves of degree 2 as calculated in proposition 2.2.2.
Since they lie on the 2-uple embedding of P 5 in P 20 they are the images of lines in P 5 .
Since dim W * = 8 we have This gives the above multiplicities. Applying these equations to the complex above yields the desired resolution.
6.2.3. Theorem. The scheme Z of last scrollar syzygies of a general canonical curve C ⊂ P 7 of genus 8 is a configuration of 14 skew conics that lie on a 2-uple embedded P 5 ֒→ P 20 . Z spans the whole P 20 of second syzygies of C.
6.2.4. Remark. Notice that h 1 (I Z/P 5 (2)) does not vanish. Therefore Z is not linearly normal.