On linear systems of \(\mathbb P^3\) through multiple points. (English) Zbl 1113.14036
Fix an integer \(d>0\), at most \(8\) general points of \(\mathbb {P}^3\) and consider the linear system \(M\) of all degree \(d\) surfaces with prescribed multiplicities at these points. Here the authors compute \(\dim (M)\) in the following way. They apply a space cubic Cremona transformation to transform \(M\) into another a similar problem \(N\) in standard form. They show that \(\dim (M)=\dim (N)\) and they give an explicit formula for \(\dim (N)\) in terms of the degree and the multiplicities needed to define \(N\).
Reviewer: Edoardo Ballico (Povo)
MSC:
| 14N05 | Projective techniques in algebraic geometry |
| 14C20 | Divisors, linear systems, invertible sheaves |
| 14J30 | \(3\)-folds |
References:
| [1] | De Volder, Cindy; Laface, Antonio, A note on the very ampleness of complete linear systems on blowings-up of \(P^3\), (Projective Varieties with Unexpected Properties (2005), de Gruyter: de Gruyter Berlin), 231-236 · Zbl 1101.14009 |
| [2] | Harbourne, Brian, The geometry of rational surfaces and Hilbert function of points in the plane, Can. Math. Soc. Conf. Proc., 6, 95-111 (1986) · Zbl 0611.14002 |
| [3] | Hirschowitz, André, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationnelles génériques, J. Reine Angew. Math., 397, 208-213 (1989) · Zbl 0686.14013 |
| [4] | Laface, Antonio; Ugaglia, Luca, On a class of special linear systems of \(P^3\), Trans. Amer. Math. Soc., 358, 12, 5485-5500 (2006), (electronic) · Zbl 1160.14003 |
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