Construction and resolutions of certain projectively normal curves
The authors give a construction for certain projectively normal \((p.n.)\) curves in \(\mathbb{P}^n(k)\), \(k\) an algebraically closed field of characteristic 0, which also furnishes a resolution of the homogeneous ideal of the curve itself. They start, following \textit{D. B. Jaffe} [Manuscr. Math. 73, No. 2, 187-205 (1991; Zbl 0779.14019)], from a \(p.n.\) curve \(C\subset \mathbb{P}^r(k)\), \(r\geq 2\), construct a cone \(\Lambda\subset\mathbb{P}^{r+1}\) over \(C\) and take a (smooth irreducible) curve \(C_m\) passing through the vertex \(v\) of \(\Lambda\) and meeting the lines of \(\Lambda\) \(m\) times outside \(v\). -- The main result of the paper can be stated as follows: All smooth irreducible curves on \(\Lambda\) are \(p.n.\), in particular if \({\mathcal J}\) is the ideal sheaf of \(C\) and \(0\to{\mathcal F}_{r-1}\to {\mathcal F}_{r-2} \to\cdots\to {\mathcal F}_1\to{\mathcal J} \to 0\) is a resolution, where \({\mathcal F}_i= \bigoplus^{\beta_i}_{j=1}{\mathcal O}_{\mathbb{P}^r}(-\alpha_{i,j})\) \(i=1,\dots,r-1\), then for all \(m\geq\max\{ \alpha_{i,j}\}-r\) there exists a smooth irreducible curve \(C_m\) on \(\Lambda\) which is \(p.n.\) The authors also give a resolution (maybe non-minimal) of the ideal sheaf \({\mathcal J}_{C_m} \subset{\mathcal O}_{\mathbb{P}^{r+1}}\) and some examples.
resolution of homogeneous ideal of curve
scientific article; zbMATH DE number 1308309
scientific article