Interpolation for curves of large degree
Let \(C\) be a nonsingular projective curve of genus \(g\) embedded by a complete linear system of a line bundle \(L\) into a projective space \(\mathbb P(H^0(L)).\) For each \(k\geq 0\) denote by \(P^k(L)\) the bundle of \(k\)-the order principal parts of \(L\). When \(k\leq \deg L -2g\) lifting global sections induces a short exact sequence \[ 0 \rightarrow R^k(L) \rightarrow H^0(L) \otimes \mathcal O_C \rightarrow P^k(L) \rightarrow 0, \] where \(R^k(L) \) is called the \(k\)-th higher order conormal bundle of \(C\). Let \(E\) be a vector bundle on \(C\) and suppose that \(h^0(E) = q \cdot \mathrm{rk} (E)+t\) for some \(0\leq t < \mathrm{rk} E\). \(E\) is said to satisfy interpolation if there exit \(q+1\) distinct points \(x_1,\dots,x_q,x\) and a vector subspace \(V \subset E|_x\) of codimension \(t\) such that the restriction morphism \(H^0(E) \rightarrow E|_S \oplus E| x/V\) is surjective, with \(S=x_1+\cdots +x_q.\) The main result of the paper under review is the following: Let \(L\) be a very ample line bundle on a nonsingular projective curve \(C\) of genus \(g\geq 0\). If there is a nonnegative integer \(k\) such that \(\deg L \geq ( k^2+k+2)g+k\), then the vector bundle \(R^k (L)^* \otimes L\) satisfies interpolation. The authors view this result as an evidence toward the validity of the conjecture of Ein--Lazarsfeld which predicts semistability of \(R^k(L)\) when the degree of \(L\) is large enough.
scientific article; zbMATH DE number 6890458