Skew configurations of lines in real del Pezzo surfaces

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2014

Öğrenci: REMZİYE ARZU ZABUN

Danışman: SERGEY FİNASHİN

Özet:

By blowing up projective plane at n<9 points which form a generic configuration, we obtain a del Pezzo surface X of degree d=9-n with a configuration of n skew lines that are exceptional curves over the blown-up points. The anti-canonical linear system maps X to a projective space of dimension d, P^{d}, and the images of these exceptional curves form a configuration of n lines in P^{d}. The subject of our research is the correspondence between the configurations of n generic points in a real projective plane, RP^{2}, and the configurations of n lines in a (9-n)-dimensional real projective space, i.e. RP^{9-n}. This correspondence is nontrivial in the cases n=6 and n=7. We study the correspondence between on such generic configurations of n points in RP^{2} and such configurations of lines in RP^{9-n}. In the case of n=6, there exist precisely 4 deformation classes of generic planar configurations of 6 points, and we describe the corresponding 4 deformation classes of configurations of 6 skew lines in RP^{3}. In the case n=7, there exist precisely 14 deformation classes of generic planar configurations of 7 points and we describe the corresponding 14 deformation classes of configurations of 7 bitangents to a quartic curve (such configurations are known in the literature as Aronhold sets).