Interpolation (Q2918459)

The paper is an overview about the interpolation problem in algebraic geometry and mainly concerns with the cases of reduced schemes and fat points.NEWLINENEWLINEIn the chapter about the reduced schemes, the author mentions the following conjecture, which is a generalization of Castelnuovo's Theorem:NEWLINENEWLINEConjecture. For \(\alpha=1,2, \dots, r-1\), if \(\Gamma \subset\mathbb{P}^r\) is a collection of \(n \geq 2r+2\alpha+1\) points in uniform position, and \(h_{\Gamma}(2) \leq 2r+\alpha\), then \(\Gamma\) is contained in a curve \(C \subset \mathbb{P}^r\) of degree at most \(r-1+\alpha\). Several evidences about this conjecture are shown. Moreover, the author shows how a proof of the conjecture would yield a complete answer to the classical problem:NEWLINENEWLINEFor which triples \((n,d,g)\) does there exist a smooth, irreducible, nondegenerate curve \(C \subset \mathbb{P}^r\) of degree \(d\) and genus \(g\) ?NEWLINENEWLINEIn the chapter about fat points, the author introduces the famous Harboune-Hirschowitz Conjecture and important remarks about it. In particular the author states the following:NEWLINENEWLINEConjecture. Let \(S\) be a general blow-up of the plane, \(C \subset S\) any integral curve. Then \((C \cdot C) \geq -1\), and if equality holds then \(C\) us a smooth rational curve.NEWLINENEWLINEThe author states that if, we believe this, the Harbourne-Hirschowitz conjecture should be equivalent to this weaker version:NEWLINENEWLINEConjecture. Let \(S\) be the blow-up of \(\mathbb{P}^2\) at general points, \(L\) any line bundle on \(S\). If the linear system \(|L|\) contains an integral curve, then \(h^1(L)=0\). \vskip0.3cmNEWLINENEWLINEFinally, the author also mentions the Alexander-Hirschowitz Theorem for double points and several results and methods (as Ciliberto-Miranda degenerations) for interpolation in algebraic geometry.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14002].