Equations and syzygies of the first secant variety to a smooth curve

We show that if C is a linearly normal smooth curve embedded by a line bundle of degree at least 2g+3+p then the secant variety to the curve satisfies N_{3,p}.


Introduction
We work throughout over an algebraically closed field k of characteristic zero. If X ⊂ P n is a smooth variety, then we let Σ i (X) (or just Σ i if the context is clear) denote the (complete) variety of (i+1)-secant i-planes. Though secant varieties are a classical subject, the majority of the work done involves determining the dimensions of secant varieties to well-known varieties. Perhaps the two most well-known results in this direction are the solution by Alexander and Hirschowitz (completed in [1]) of the Waring problem for homogeneous polynomials and the classification of the Severi varieties by Zak [26].
It was conjectured in [13] and it was shown in [28] that if C is a smooth curve embedded by a line bundle of degree at least 4g + 2k + 3 then Σ k is set theoretically defined by the (k + 2) × (k + 2) minors of a matrix of linear forms. This was extended in [17] where the degree bound was improved to 4g + 2k + 2 and it was shown that the secant varieties are scheme theoretically cut out by the minors. It was further shown in [34] that if X ⊂ P n satisfies condition N 2 then Σ 1 (v d (X)) is set theoretically defined by cubics for d ≥ 2.
In [35] it was shown that if C is a smooth curve embedded by a line bundle of degree at least 2g + 3 then I Σ 1 is 5-regular, and under the same hypothesis it was shown in [30]  analogous well-known facts for the curve C itself ( [15], [18], [27]) this led to the following conjecture, extending that found in [34]: Conjecture 1.1. [30] Suppose that C ⊂ P n is a smooth linearly normal curve of degree d ≥ 2g + 2k + 1 + p, where p, k ≥ 0. Then (1) Σ k is ACM and I Σ k has regularity 2k + 3 unless g = 0, in which case the regularity is k + 2.
Remark 1.2. Recall [12] that a variety Z ⊂ P n satisfies N r,p if the ideal of Z is generated in degree r and the syzygies among the equations are linear for p − 1 steps. Note that the better-known condition N p [18] implies N 2,p .
By the work of Green and Lazarsfeld [18], [25], the conjecture holds for k = 0. Further, by [14] and by [37] it holds for g ≤ 1, and by [30] parts (1) and (2) hold for k = 1. In this work, we show that part (3) holds for k = 1 (Theorem 3.5). Some analogous results for higher dimensional varieties can be found in [36].
Our approach combines the geometric knowledge of secant varieties mentioned above with the well-known Koszul approach of Green and Lazarsfeld. To fix notation, if L is a vector bundle on a smooth curve C, then we let where d : C × C → S 2 C is the natural double cover, and if F is a globally generated coherent sheaf on a variety X, then we have the coherent sheaf M F defined via the exact sequence 0 → M F → Γ(X, F) → F → 0. As we will be interested only in the first secant variety for the remainder of the paper, we write Σ for Σ 1 .

Preliminaries
Our starting point is the familiar: Proposition 2.1. Let C ⊂ P n be a smooth curve embedded by a line bundle L of degree at least 2g + 3. Then Σ satisfies N 3,p if and only if Proof: Because L also induces an embedding Σ ⊂ P n , we abuse notation and denote the associated vector bundle on Σ by M L . Letting F = ⊕Γ(Σ, O Σ (n)) and applying [11, 5.8] to O Σ gives the exact sequence: As Σ is ACM [30], the term on the right vanishes.
y y t t t t t t t t t t t • π is the blow up of Σ along C • i is the inclusion of the exceptional divisor of the blow-up • d is the double cover, π i are the projections • ϕ is the morphism induced by the linear system |2H − E| which gives Σ the structure of a P 1 -bundle over S 2 C; note in particular that Σ is smooth. We make frequent use of the rank 2 vector bundle Proof: This follows immediately from the 5-term sequence associated to the Leray-Serre spectral sequence: and Proposition 2.1.
We will need a cohomological result: Lemma 2.4. Let C ⊂ P n be a smooth curve embedded by a line bundle L with deg(L) ≥ 2g + 3.

Main Result
We first reinterpret the injection in Proposition 2.3 as a vanishing on Σ (Proposition 3.1), then on S 2 C (Corollary 3.3), and finally on C × C (Theorem 3.5).
Proposition 3.1. Let C ⊂ P n be a smooth curve satisfying N p embedded by a line bundle L with deg(L) ≥ 2g + 3.
Proof: We use Proposition 2.3. Consider the sequence on Σ We know The first equality follows as the restriction of π * ∧ a M L (bH) to Z is ∧ a M L (bH)⊠O C . For the second we use the Künneth formula together with the fact that h 1 (C, ∧ a M L ⊗ L b ) = 0 as C satisfies N p [18]. The third is the last part of 2.2. Thus and so by Proposition 2.3 it is enough to show that H 1 ( Σ, π * ∧ a M L ⊗O(bH −E)) = 0 for 2 ≤ a ≤ p + 1, b ≥ 2. From the sequence Lemma 3.2. Let C ⊂ P n be a smooth curve embedded by a line bundle L with deg(L) ≥ 2g + 3 and consider the morphism ϕ : Σ → S 2 C ⊂ P s induced by the Proof: Consider the diagram on Σ: The vertical map in the middle is surjective as we have Γ( L). Therefore, surjectivity of the lower right horizontal map and commutativity of the diagram show that the righthand vertical map is surjective.
Σ by the projection formula and that the higher direct image sheaves R i ϕ * O e Σ vanish as Σ is a P 1 -bundle over S 2 C. For the higher direct images, we have R i ϕ * π * L = 0 as the restriction of L to a fiber of ϕ is O(1) and hence the cohomology along the fibers vanishes. From the rightmost column, we see R i ϕ * K = 0. From the leftmost column, we have the sequence Combining Proposition 3.1 with Lemma 3.2 yields: Corollary 3.3. Let C ⊂ P n be a smooth curve satisfying N p embedded by a line bundle L with deg(L) ≥ 2g + 3. Then Σ satisfies N 3,p if We need a technical lemma, completely analogous to [25, 1.4.1].
Lemma 3.4. Let X ⊂ P n be a smooth curve embedded by a non-special line bundle L satisfying N 2,2 , let x 1 , · · · , x n−2 be a general collection of distinct points, and let D = x 1 + · · · + x n−2 . Then there is an exact sequence of vector bundles on X × X Proof: Choose a general point x 1 ∈ X and consider the following diagram on X × X: where the center column comes from [25, 1.4.1]. Following just as in that proof, we obtain Proof: First note [18] that such a curve satisfies N p+2 . We verify the condition in Corollary 3.3. Pulling the sequence on On the right, we have a direct sum of vector bundles of the form F ⊠ F (−∆) where F is a line bundle of degree deg(L) − r. Thus H 1 and H 2 of the right side will vanish when deg(L) − r ≥ 2g + 1.
On the left, we have a direct sum of vector bundles of the form F ⊠ F where F is a line bundle of degree n − 2 − (r − 1) = deg(L) − g − r − 1. Because x 1 , · · · , x n−2 are general, H 1 and H 2 of the left side will vanish when deg(L) − g − r − 1 ≥ g.

Acknowledgments
This project grew out of work done together with Jessica Sidman, and has benefited greatly from her insight and input, as well as from her comments regarding preliminary drafts of this work.