Syzygies of secant varieties of curves of genus 2

Ein, Niu and Park showed in [ENP20] that if the degree of the line bundle $L$ on a curve of genus $g$ is at least $2g+2k+1$, the $k$-th secant variety of the curve via the embedding defined by the complete linear system of $L$ is normal, projectively normal and arithmetically Cohen-Macaulay, and they also proved some vanishing of the Betti diagrams. However, the length of the linear strand of weight $k+1$ of the resolution of the secant variety $\Sigma_k$ of a curve of $g\geq2$ is still mysterious. In this paper we calculate the complete Betti diagrams of the secant varieties of curves of genus $2$ using Boij-S\"{o}derberg theory. The main idea is to find the pure diagrams that contribute to the Betti diagram of the secant variety via calculating some special positions of the Betti diagram.


Introduction
Let C be a smooth projective curve of genus g over the complex field C. Let L be a very ample line bundle on C such that its complete linear system defines an embedding C ֒→ P r .For an integer k ≥ 0, define the k-th secant variety Σ k = Σ k (C, L) ⊂ P r to be the Zariski closure of the union of all (k + 1)-secant k-planes to C in P r .Formally, the secant varieties can be realized as images of the projectivizations of secant bundles.Concretely, let pr : C × C k → C be the projection on the first factor, where C k denotes the k-th symmetric product of C. We have the canonical morphism σ : C × C k → C k+1 which sends (x, ξ) to x + ξ.Then the (k + 1)-th secant sheaf E k+1,L or simply E L is defined to be the rank k + 1 vector bundle σ * pr * L. The projective bundle is defined to be the projectivization B k (L) := P(E L ).Let O B k (1) := O P(E L ) (1) be the tautological bundle on B k .It induces a morphism β : B k → PH 0 (C k+1 , E L ) = P r .The k-th secant variety Σ k can be realized as the image β.We may assume that r ≥ 2k + 3, because by [Lan84] the secant varieties of curves always have the expected dimension, meaning that for Σ k ⊂ P r , we have dim Σ k = min{2k + 1, r}.Concretely, Σ k = P r if r ≤ 2k + 1 and Σ k is a hypersurface with degree calculated in Proposition 5.10 of [ENP20] if r = 2k + 2. Let O Σ k (1) be the line bundle on Σ k that is the pullback of O P r (1).
We are interested in the syzygies of Σ k .To that end, we introduce some notations.Let X be a projective variety embedded in P r = PH 0 (X, L) via a very ample line bundle L. Let S = SymH 0 (X, L) and R = m≥0 H 0 (X, B ⊗ mL) viewed as a graded S-module, where B is a coherent sheaf on X.By Hilbert's syzygy theorem, there is a minimal graded free resolution of R over S: where E p is a free S-module.We define the Koszul cohomology group K p,q (X, B; L) to be the suitable C-linear space such that E p = q K p,q (X, B; L) ⊗ C S(−p − q).
The Koszul cohomology group K p,q (X, B; L) can be identified with the homology of the following differentials: where the morphism for arbitrary p, q ∈ Z.If B = O X the structure sheaf on X, the Koszul cohomology groups are simply denoted as K p,q (X, L).We have the Betti diagram of which the column p and row q is dim K In this paper, we will focus on curves of genus 2 when the kernel bundle of the evaluation map of the canonical bundle on the curve is a line bundle, and thus easier to describe.We assume that r ≥ 2k + 3, which means that deg(L) ≥ 2k + 5.It was shown from [ENP20] Furthermore, by Danila's theorem(see [Dan07]), we have dim K i,j (Σ k , O Σ k (1)) = 0 for j ≤ k, except from (i, j) = (0, 0), for which we have dim K 0,0 (Σ k , O Σ k (1)) = 1.We will prove the followings: Based on the values calculated above, we can determine the length of the linear strand together with the unknown Koszul cohomology group K r−2k−1,2k+1 .
Theorem 1.2.Let Σ k be the the secant variety of a genus 2 curve. ( (4)For k = 0 and 1 ≤ i ≤ r − 2, we have Summarizing the results above, we deduce that the Betti diagram of Σ k (k ≥ 1) is of the following shape: where the blanks are 0 and the asterisks stand for non-zero terms, the values of which were shown in Theorem 1.2 (3).

The vanishing of K r−2k−1,2k+1
Before proving the main theorem in this section, it is useful to introduce the duality property of Koszul cohomology.
Proposition 2.1.Let L be a globally generated line bundle on a smooth projective scheme X of dimension n and B an arbitrary line bundle on X.
Proof.The proof is essentially the same as that of Theorem 2.25 of [AN10].However, the smoothness assumed there is not necessary because the ingredient is the Serre's duality.
From now on, we assume that C is a smooth projective curve of genus 2 over C and L is a line bundle of degree at least 2k + 5.The complete system of L defines an embedding C ֒→ P r with r ≥ 2k + 3. Let Σ k ⊂ P r be the k-th secant variety of C. Since all intermediate cohomology H i (Σ k , O Σ k (l)) with l ∈ Z and 1 ≤ i ≤ 2k vanish by Theorem 5.2 and Theorem 5.8 of [ENP20], the duality property holds.In other words, we have where ω Σ k is the canonical sheaf on Σ k .Therefore to show the vanishing of this group, it is equivalent to show the following: Theorem 2.2.The morphism is surjective with the assumption r ≥ 2k + 3.
Proof.By the construction of the secant varieties and Theorem 5.8 of [ENP20], we know that where K is the canonical divisor on C and the morphism φ becomes given by We still denote it by φ and assume J = Im(φ).Write H 0 (C, K) = span{1, v}, where 1 and v are seen as rational functions on an affine open subset of C.
From now on I omit the symbol C for short.As long as the multiplication morphism Once we have shown this, we would have vx We prove it by induction on m.

and we have vy
Assume that we have reduced the problem to show v p x p ⊗ v m ∈ J for all x p ∈ H 0 (L − pK).If we further assume that ψ p+1 : and the latter is in J by the induction hypothesis.We are left to show v p+1 x p+1 ⊗ v m ∈ J.We do the step above for k times.Then we are left to show For the deduction above, we need the surjectivity of the morphisms To verify the desired surjectivity, it suffices to check Proof.From the duality property we have where the latter group is the homology of the sequence We have H 0 (ω Σ k (−1)) = H 2k+1 (O Σ k (1)) ∨ = 0 by Theorem 5.2 of [ENP20].By Theorem 2.2, the map on the right is surjective, Therefore 3 Boij-Söderberg theory In this section we recall some useful facts from Boij-Söderberg theory.One may refer to [ES09] and [Tay22] for this part.Next we introduce the multiplicity of a module and extend it to the formal diagrams.
Definition 3.4.Let S be as above and M a graded S-module.The Hilbert series HS M (t) = j dim M j t j of M can be uniquely represented as Define the multiplicity of M to be HN M (1).
From Corollary of [HK84] and Theorem 1.2 of [HM85], we know that Theorem 3.5.For a pure resolution Recall that the Hilbert functions of a module are totally determined linearly by the Betti numbers(Corollary 1.10 of [Eis05]).Therefore they can be generalized to formal diagrams.Furthermore, the multiplicity of a formal diagram can be defined.The multiplicity should be also totally determined linearly by the Betti numbers.So we can calculate the multiplicities of the pure diagrams defined in Definition 3.2.
Proposition 3.6.The multiplicities of the pure diagrams defined in Definition 3.2 are 1.
Proof.We first check that when normalized such that dim K 0,0 = 1, the Betti numbers of the pure diagrams satisfy the conditions in Theorem 3.5.In fact, we have Since the multiplicity is totally determined by the Betti numbers, and the Betti numbers of the pure diagrams divided by dim K 0,0 satisfy the conditions in Theorem 3.5, the result of Theorem 3.5 is formally generalized.Then the multiplicity of β(e) dim K 0,0 is

The complete Betti diagram of Σ k
We first show that the multiplicity of a projective variety coincides with its degree.
Proposition 4.1.Let V ⊂ P r be a projective variety.Let S = C[x 0 , • • • , x r ] and I be the vanishing ideal of V .Then the multiplicity of S/I coincides with the degree of V .Here the degree means the number of points in the intersections , where H i 's are general hyperplanes in P r .
Proof.The Hilbert series HS Taking the difference of these two equations, we get (1 Here the symbol ∼ = means that they are equal from a term of sufficiently large degree.Inductively, we get Dt j , where D = deg(V ).Multiplied by (1 − t) on both sides, it is deduced that By the definition of the multiplicity, we see that it is equal to the degree.
We fix some notations.Observe that for the Betti diagram of Σ k , dim K 1,k+1 = 0 and the lower right corner is (r − 2k − 1, 2k + 2).These mean that the sequence e that characterizes the diagram β(e) composing the Betti diagram of Σ k has the first non-zero entry starting from k + 2, and the last entry at most r + 1.
From Theorem 1.3 of [SV09] we know that the linear strand of weight k + 1 of Σ k has length at least r − 2k − 3.This implies that k Now we can prove our main theorem: Theorem 4.3.Assume that k ≥ 1.Then for the secant variety Σ k of a genus 2 curve, we have: Proof.Consider K r−2k−1,2k+2 .For a pure diagram characterised by e contributing to K r−2k−1,2k+2 , we have e r−2k−1 = r − 2k − 1 + 2k + 2 = r + 1.This implies that only those i with i 0 > 0 contribute to K r−2k−1,2k+2 .Applying Lemma 4.2 to K r−2k−1,2k+2 , we have (1) Since we have It can be deduced from Proposition 5.10 of [ENP20] that, when k ≥ 1, we have Therefore the equation (1) becomes Consider K r−2k−2,2k+2 .For the pure diagrams characterised by e contributing to K r−2k−2,2k+2 , we have e r−2k−2 = r − 2k − 2 + 2k + 2 = r, which implies that e r−2k−1 = r + 1 Since we have As a result, We now plug the value of c (2,••• ,2);d in the equation (2): Moving the term with (k + 2)! to the left hand side and canceling the factor For the right hand side, note that for each i, we have i 0 = 1 and i j ≤ 2 for 1 ≤ j ≤ k.Therefore Applying this inequality to (3), we know that The inequalities are actually equalities!This implies, firstly, only the coefficient of (1, 2, • • • , 2) is non-zero among i with i 0 = 1, and secondly, The degree sequence associated to (2, Moreover, both of these pure diagrams have linear strands of length r − 2k − 3.This finishes the proof.
Corollary 4.4.With the same notations as Theorem 4.3, for 1 ≤ i ≤ r − 2k − 3, we have Proof.By the definition of pure diagrams, we have Plugging them in dim K i,k+1 , we get For the case k = 0, the secant variety is just the curve itself.The problem about the length of linear strand is the Green-Lazarsfeld Conjecture and was solved in [EL15].But I would like to calculate it again using Boij-Söderberg theory, which can give accurate values.
Plugging these into dim K i,1 , we get At the end of this paper, I will prove a complemental result of N k+2,r−2k−3 .We can observe that K i,2k+2 (Σ k , O Σ k (1)) = 0 if and only if r − 2k − 2 ≤ i ≤ r − 2k − 1.For curves of genus 2, this can be easily seen from the shapes of pure diagrams contributing to the Betti diagrams of secant varieties, but I would prove a more general result for curves of general genus.Proposition 4.6.Let C be a curve of genus g ≥ 1.Let L be a very ample line bundle with deg(L) ≥ 2g + 2k + 1 and hence H 0 (L − K) = 0. Assume that the embedding C ֒→ P r is defined by the complete linear system of L. Let Σ k be the k-th order secant variety of C. Then Proof.The implication from left to right was proved in Theorem 5.2 (4) of [ENP20].Now assume that r − g − 2k ≤ i ≤ r − 1 − 2k.We want to show K i,2k+2 (Σ k , O Σ k (1)) = 0. We essentially use the ideas in the proof of Proposition 5.1 of [EL12].Notice that we have the duality K i,2k+2 (Σ k , O Σ k (1)) ∼ = K r−(2k+1)−i,0 (Σ k , ω Σ k ; O Σ k (1)) ∨ .Essentially we have to show the morphism is not injective for 0 ≤ i ≤ g − 1.We are interested in the case i ≥ 1.Take f 0 , • • • , f i linearly independent sections of H 0 (C, K C ) and s ∈ H 0 (C, L − K).Then ∀F ∈ S k H 0 (C, K C ) nonzero, the element is mapped to 0.Here we identify H 0 (Σ k , ω Σ k ) with S k+1 H 0 (C, K C ) and H 0 (Σ k , O Σ k (1)) with H 0 (C, L).
Definition 3.1.The rational vector space B = ∞ −∞ Q n+1 is called the space of rational Betti diagrams with n + 1 columns and rows numbered by integers.Definition 3.2.A pure diagram β(e) is the diagram characterised by a degree sequence e = (e 0 , • • • , e n ) ∈ Z n+1 with e 0 < e 1 < • • • < e n , for which the Betti numbers are defined by dim K p,q (β(e)i − e p | , p + q = e p 0, else From Theorem 0.1 and Theorem 0.2 of [ES09], we can get the theorem of decompositions of the Betti diagram of a Cohen-Macaulay module into combinations of pure diagrams.Theorem 3.3.Let S be the polynomial ring C[x 0 , • • • , x r ].For any finitely generated Cohen-Macaulay graded S-module M , its Betti diagram β(M ) can be (not necessarily uniquely) decomposed as e c e β(e) with c e ≥ 0.
where R is a graded S-module and in fact Cohen-Macaulay with the numerical conditions of b i , the multiplicity of R is µ(R) = 2);d .In other words, the Betti diagram of Σ k is decomposed into the combination of 2 pure diagrams, which are represented by (2, • • • , 2) and (1, 2, • • • , 2).