Regularity bounds by minimal generators and Hilbert function
Let \(I_X \subset S = K[x_0,\dots,x_n]\) (where \(K\) is an algebraically closed field) be a saturated homogeneous ideal defining a closed subscheme \(X \subset \mathbb P^n\). The (Castelnuovo-Mumford) regularity of \(I_X\) (or of \(X\)), \(\mathrm{reg}(I_X)\), is the smallest integer \(m\) such that for all \(i\), the \(i\)-th syzygy module of \(I_X\) is generated in degree \(\leq m+i\). Recall that \(\mathrm{reg}(I_X)\) is an upper bound for the degrees of the minimal generators of \(I_X\). Also associated to \(I_X\) is the Hilbert function \(H_X (t) := \dim_K [S/I_X]_t\). For \(t \gg 0\), \(H_X (t)\) coincides with the Hilbert polynomial \(P_X (t)\). The regularity of \(H_X \) is defined to be \(\rho_X := \min \{ \bar t \;| \;H_X (t) = P_X (t)\) for all \(t \geq \bar t \}\). From now on we assume consider a curve \(C\) in \(\mathbb P^n\). Let \(\beta_1,\dots,\beta_{n-1}\) be degrees for which there exists a complete intersection of type \((\beta_1,\dots,\beta_{n-1})\) containing \(C\). Then \(\mathrm{reg}(I_C)\) is bounded above by \(\max \{ \rho_C +1, \beta_1 + \cdots + \beta_{n-1} - (n-1) \}\). Furthermore, the authors show that this bound is sharp if \(C\) is directly linked to a curve \(Y\) with degenerate general hyperplane section. In characteristic 0, the authors also show the following. Let \(C \subset \mathbb P^3\) be a non-degenerate curve with \(\mathrm{reg}(C) > \rho_C +1\), and assume that \(\beta_1, \beta_2\) (as above) are the smallest possible. Then \(C\) is directly linked to a plane curve \(C'\) by a complete intersection of type \((\beta_1,\beta_2)\) if and only if \(\mathrm{reg}(C) = \beta_1 + \beta_2 -2\). The authors end the paper with a number of examples.
scientific article; zbMATH DE number 5563710
scientific article