We prove that the Segre-Gimigliano-Harbourne-Hirschowitz conjecture holds for quasi-homogeneous linear systems on ℙ² for m = 7, 8, 9, 10, i.e. systems of curves of a given degree passing through points in general position with multiplicities at least m,...,m,m₀, where m = 7, 8, 9, 10, m₀ is arbitrary.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap100-3-5,
author = {Marcin Dumnicki},
title = {Quasi-homogeneous linear systems on P2 with base points of multiplicity 7, 8, 9, 10},
journal = {Annales Polonici Mathematici},
volume = {101},
year = {2011},
pages = {277-300},
zbl = {1219.14042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap100-3-5}
}
Marcin Dumnicki. Quasi-homogeneous linear systems on ℙ² with base points of multiplicity 7, 8, 9, 10. Annales Polonici Mathematici, Tome 101 (2011) pp. 277-300. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap100-3-5/