Rendiconti del Seminario Matematico della Università di Padova


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Volume 136, 2016, pp. 191–203
DOI: 10.4171/RSMUP/136-13

On the Sylvester–Gallai theorem for conics

Adam Czapliński[1], Marcin Dumnicki[2], Łucja Farnik[3], Janusz Gwoździewicz[4], Magdalena Lampa-Baczyńska[5], Grzegorz Malara[6], Tomasz Szemberg[7], Justyna Szpond[8] and Halszka Tutaj-Gasińska[9]

(1) Institut für Mathematik, Johannes Gutenberg-Universität Mainz, 55099, Mainz, Germany
(2) Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348, Kraków, Poland
(3) Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348, Kraków, Poland
(4) Department of Mathematics, Pedagogical University of Cracow, ul. Podchorążych 2, 30-084, Kraków, Poland
(5) Department of Mathematics, Pedagogical University of Cracow, ul. Podchorążych 2, 30-084, Kraków, Poland
(6) Department of Mathematics, Pedagogical University of Cracow, ul. Podchorążych 2, 30-084, Kraków, Poland
(7) Department of Mathematics, Krakow Pedagogical Academy, ul. Podchorążych 2, 30-084, Kraków, Poland
(8) Department of Mathematics, Pedagogical University of Cracow, ul. Podchorążych 2, 30-084, Kraków, Poland
(9) Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348, Kraków, Poland

In the present note we give a new proof of a result due to Wiseman and Wilson [13] which establishes an analogue of the Sylvester–Gallai theorem valid for curves of degree two. The main ingredients of the proof come from algebraic geometry. Specically, we use Cremona transformation of the projective plane and Hirzebruch inequality (1).

Keywords: Arrangements of subvarieties, combinatorial arrangements, Sylvester–Gallai problem, Cremona transformation, Hirzebruch inequality, interpolation problem

Czapliński Adam, Dumnicki Marcin, Farnik Łucja, Gwoździewicz Janusz, Lampa-Baczyńska Magdalena, Malara Grzegorz, Szemberg Tomasz, Szpond Justyna, Tutaj-Gasińska Halszka: On the Sylvester–Gallai theorem for conics. Rend. Sem. Mat. Univ. Padova 136 (2016), 191-203. doi: 10.4171/RSMUP/136-13