# Rendiconti del Seminario Matematico della Università di Padova

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. Specically, 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