In this paper, we show that for a nonsingular projective curve and a positive integer k, the k-th secant bundle is the blowup of the k-th secant variety along the (k−1)-th secant variety. This answers a question raised in the recent paper of the authors on secant varieties of curves.
Citation: Lawrence Ein, Wenbo Niu, Jinhyung Park. On blowup of secant varieties of curves[J]. Electronic Research Archive, 2021, 29(6): 3649-3654. doi: 10.3934/era.2021055
[1] | Lawrence Ein, Wenbo Niu, Jinhyung Park . On blowup of secant varieties of curves. Electronic Research Archive, 2021, 29(6): 3649-3654. doi: 10.3934/era.2021055 |
[2] | Shishuo Fu, Jiaxi Lu, Yuanzhe Ding . A skeleton model to enumerate standard puzzle sequences. Electronic Research Archive, 2022, 30(1): 179-203. doi: 10.3934/era.2022010 |
[3] | Lie Fu, Victoria Hoskins, Simon Pepin Lehalleur . Motives of moduli spaces of rank $ 3 $ vector bundles and Higgs bundles on a curve. Electronic Research Archive, 2022, 30(1): 66-89. doi: 10.3934/era.2022004 |
[4] | Liu Yang, Yuehuan Zhu . Second main theorem for holomorphic curves on annuli with arbitrary families of hypersurfaces. Electronic Research Archive, 2024, 32(2): 1365-1379. doi: 10.3934/era.2024063 |
[5] | Álvaro Antón-Sancho . A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves. Electronic Research Archive, 2024, 32(11): 6109-6119. doi: 10.3934/era.2024283 |
[6] | Fabrizio Catanese, Luca Cesarano . Canonical maps of general hypersurfaces in Abelian varieties. Electronic Research Archive, 2021, 29(6): 4315-4325. doi: 10.3934/era.2021087 |
[7] | Xin Lu, Jinbang Yang, Kang Zuo . Strict Arakelov inequality for a family of varieties of general type. Electronic Research Archive, 2022, 30(7): 2643-2662. doi: 10.3934/era.2022135 |
[8] | Yuri Prokhorov . Conic bundle structures on $ \mathbb{Q} $-Fano threefolds. Electronic Research Archive, 2022, 30(5): 1881-1897. doi: 10.3934/era.2022095 |
[9] | Yitian Wang, Xiaoping Liu, Yuxuan Chen . Semilinear pseudo-parabolic equations on manifolds with conical singularities. Electronic Research Archive, 2021, 29(6): 3687-3720. doi: 10.3934/era.2021057 |
[10] | Miguel Ferreira, Luís Moreira, António Lopes . Differential drive kinematics and odometry for a mobile robot using TwinCAT. Electronic Research Archive, 2023, 31(4): 1789-1803. doi: 10.3934/era.2023092 |
In this paper, we show that for a nonsingular projective curve and a positive integer k, the k-th secant bundle is the blowup of the k-th secant variety along the (k−1)-th secant variety. This answers a question raised in the recent paper of the authors on secant varieties of curves.
Throughout the paper, we work over an algebraically closed field
Σk=Σk(C,L)⊆Pr |
of
Assume that
σk+1:Ck×C⟶Ck+1 |
be the morphism sending
Ek+1,L:=σk+1,∗p∗L, |
which is a locally free sheaf of rank
Bk(L):=P(Ek+1,L) |
equipped with the natural projection
H0(Bk(L),OBk(L)(1))=H0(Ck+1,Ek+1,)=H0(C,L), |
and therefore, the complete linear system
βk:Bk(L)⟶Pr=P(H0(C,L)). |
The
It is clear that there are natural inclusions
C=Σ0⊆Σ1⊆⋯⊆Σk−1⊆Σk⊆Pr. |
The preimage of
Theorem 1.1. Let
To prove the theorem, we utilize several line bundles defined on symmetric products of the curve. Let us recall the definitions here and refer the reader to [2] for further details. Let
Ck+1=C×⋯×C⏟k+1times |
be the
Ak+1,L:=Tk+1(L)(−2δk+1) |
be a line bundle on
The main ingredient in the proof of Theorem 1.1 is to study the positivity of the line bundle
Proposition 1.2. Let
In particular, if
In this section, we prove Theorem 1.1. We begin with showing Proposition 1.2.
Proof of Proposition 1.2. We proceed by induction on
Assume that
rz,k+1,L:H0(Ck+1,Ak+1,L)⟶H0(z,Ak+1,L|z) |
is surjective. We can choose a point
rz,k+1,L:H0(Ck+1,Ak+1,L)⟶H0(z,Ak+1,L|z) |
where all rows and columns are short exact sequences. By tensoring with
rz,k+1,L:H0(Ck+1,Ak+1,L)⟶H0(z,Ak+1,L|z) |
in which we use the fact that
Since
Lemma 2.1. Let
Proof. Note that
B′/A′⊗A′A′/m′q=B′/(m′qB′+A′)=B′/(m′p+A′)=0. |
By Nakayama lemma, we obtain
We keep using the notations used in the introduction. Recall that
αk,1:Bk−1(L)×C⟶Bk(L). |
To see it in details, we refer to [1,p.432,line –5]. We define the relative secant variety
Proposition 2.2. ([2,Proposition 3.15,Theorem 5.2,and Proposition 5.13]) Recall the situation described in the diagram
αk,1:Bk−1(L)×C⟶Bk(L). |
Let
1.
2.
3.
As a direct consequence of the above proposition, we have an identification
H0(Ck+1,Ak+1,L)=H0(Σk,IΣk−1|Σk(k+1)). |
We are now ready to give the proof of Theorem 1.1.
Proof of Theorem 1.1. Let
b:˜Σk:=BlΣk−1Σk⟶Σk |
be the blowup of
b:˜Σk:=BlΣk−1Σk⟶Σk |
We shall show that
Write
γ:˜Σk⟶P(V). |
On the other hand, one has an identification
ψ:Ck+1⟶P(V). |
Also note that
ψ:Ck+1⟶P(V). |
Take an arbitrary closed point
α−1(x)⊆π−1k(x″)∩β−1k(x′). |
However, the restriction of the morphism