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**Linear systems of plane curves with base points of equal multiplicity.**
*(English)*
Zbl 0959.14015

The authors study the following classical interpolation problem: Given a finite set of points \(\{P_1,\dots, P_n\}\) in the projective plane and a set \(\{m_1,\dots, m_n\}\) of non negative integers, determine the dimension of the linear system \(|L|\) of plane curves of degree \(d\) passing through each \(P_i\) with multiplicity greater or equal than \(m_i\).

Since every \((m_i)\)-uple point imposes \(m_i(m_i-1)/2\) conditions to curves, it is natural to expect that \(|L|\) has dimension either \((d(d-3)-\sum_1^n m_i(m_i-1))/2\) or is empty, when the previous number is negative. On the other hand it is easy to write down examples in which the true dimension is different from the expected one. Harbourne and Hirschowitz conjectured that the true and the expected dimensions differ exactly when the the proper transform of \(|L|\) in the blow up of \({\mathbb P}^2\) at the \(P_i\)’s has a fixed rational component of self intersection \(-1\). By now, only few particular cases of the conjecture are established.

The authors examine here the homogeneous case, in which all the multiplicities are the same (say equal to \(m\)). In this situation, they give an explicit description of all the cases in which the \((-1)\)-curve quoted above actually exists. Then they use an inductive procedure, which relies on the degeneration of the plane to a union of rational surfaces, to attack the conjecture. The authors are able to control the inductive steps and prove the conjecture for all \(m\leq 12\).

Since every \((m_i)\)-uple point imposes \(m_i(m_i-1)/2\) conditions to curves, it is natural to expect that \(|L|\) has dimension either \((d(d-3)-\sum_1^n m_i(m_i-1))/2\) or is empty, when the previous number is negative. On the other hand it is easy to write down examples in which the true dimension is different from the expected one. Harbourne and Hirschowitz conjectured that the true and the expected dimensions differ exactly when the the proper transform of \(|L|\) in the blow up of \({\mathbb P}^2\) at the \(P_i\)’s has a fixed rational component of self intersection \(-1\). By now, only few particular cases of the conjecture are established.

The authors examine here the homogeneous case, in which all the multiplicities are the same (say equal to \(m\)). In this situation, they give an explicit description of all the cases in which the \((-1)\)-curve quoted above actually exists. Then they use an inductive procedure, which relies on the degeneration of the plane to a union of rational surfaces, to attack the conjecture. The authors are able to control the inductive steps and prove the conjecture for all \(m\leq 12\).

Reviewer: L.Chiantini (Siena)

### MSC:

14H50 | Plane and space curves |

14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |

14C20 | Divisors, linear systems, invertible sheaves |

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\textit{C. Ciliberto} and \textit{R. Miranda}, Trans. Am. Math. Soc. 352, No. 9, 4037--4050 (2000; Zbl 0959.14015)

### References:

[1] | C. Ciliberto and R. Miranda: “Degenerations of Planar Linear Systems”, J. Reine Angew. Math. 501 (1998), 191-220. CMP 98:16 · Zbl 0943.14002 |

[2] | A. Gimigliano: “On Linear Systems of Plane Curves”. Ph.D. Thesis, Queen’s University, Kingston, Ontario, Canada (1987). · Zbl 0702.14003 |

[3] | Alessandro Gimigliano, Our thin knowledge of fat points, The Curves Seminar at Queen’s, Vol. VI (Kingston, ON, 1989) Queen’s Papers in Pure and Appl. Math., vol. 83, Queen’s Univ., Kingston, ON, 1989, pp. Exp. No. B, 50. · Zbl 0743.14005 |

[4] | Brian Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane, Proceedings of the 1984 Vancouver conference in algebraic geometry, CMS Conf. Proc., vol. 6, Amer. Math. Soc., Providence, RI, 1986, pp. 95 – 111. · Zbl 0611.14002 |

[5] | André Hirschowitz, La méthode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Math. 50 (1985), 337 – 388 (French, with English summary). · Zbl 0571.14002 |

[6] | André Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationnelles génériques, J. Reine Angew. Math. 397 (1989), 208 – 213 (French). · Zbl 0686.14013 |

[7] | M. Nagata: On rational surfaces II, Memoirs of the College of Science, University of Kyoto, Series A, Vol. 33, Mathematics No. 2, (1960), 271-293. · Zbl 0100.16801 |

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