On blowup of secant varieties of curves
In this article the authors identify the \(k\)-th secant bundle \(B^k(L)\) on a nonsingular projective curve \(C\) with the blowup of its \(k\)-th secant variety \(\Sigma_k(C,L)\) along the \((k-1)\)-th secant variety \(\Sigma_{k-1}(C,L)\). This work is a continuation of a previous article of them in which they showed that \(B^k(L)\) is the normalization of \(\Sigma_k(C,L)\) along \(\Sigma_{k-1}(C,L)\). The working setting is the following: the ground field is algebraically closed of characteristic zero; \(C\) is a nonsingular projective curve of genus \(g\geq 0\); \(L\) is a very ample line bundle on \(C\) giving the embedding \(C\hookrightarrow \mathbb P (H^0(C,L))\); \(k\geq 0\) is an integer satisfying the inequality \(\deg L \geq 2g+2k+1\). In Section 1 the authors recall the definition of the \(k\)-th secant bundle \(B^k(L)\) on \(C\) and show how to use it to give an alternative definition of \(\Sigma_k\). In this construction a key role is played by the property of ``separating points'', which \(L\) has under the above assumption on \(\deg L\): in this case, the identification \(H^0(B^k(L),\mathcal{O}_{B^k(L)}(1))=H^0(C,L)\) holds and the complete linear system \(|\mathcal{O}_{B^k(L)}(1)|\) induces a morphism \(\beta_k: B^k(L)\rightarrow \mathbb P(H^0(C,L))\). Thus \(\Sigma_k\) is defined as the image of the morphism \(\beta_k\). Moreover, since \(\beta_k^{-1}(\Sigma_{k-1})\) is a divisor on \(B^k(L)\), one gets a morphism from \(B^k(L)\) to the blowup of \(\Sigma_k\) along \(\Sigma_{k-1}\). In order to prove that \(B^k(L)\) actually is the blowup itself (Theorem 1.1), the authors define a line bundle \(A_{k+1,L}\) on the symmetric product \(C_{k+1}:=C^{\times (k+1)}/\mathfrak{S}_{k+1}\) and prove that it is very ample under the above assumption on \(\deg L\) (Proposition 1.2). In Section 2 the above results are proved together with other preliminary ones.
scientific article; zbMATH DE number 7438949