Abstract
Polarized rational surfaces (X, L) of sectional genus two ruled in conics are studied. When they are not minimal, they are described as the blow-up of 𝔽1 at some points lying on distinct fibers. Ampleness and very ampleness of L are studied in terms of their location. When L is very ample and there is a line contained in X and transverse to the fibers, the conic fibrations (X, L) are classified and a related property concerned with the inflectional locus is discussed.
Acknowledgements
The authors are grateful to the referee for useful remarks. The first author is a member of G.N.S.A.G.A. of the Italian INdAM. He would like to thank the PRIN 2015 Geometry of Algebraic Varieties and the University of Milano for partial support. The second author wants to thank the Spanish Ministry of Science, Innovation and Universities (Project MTM 2015-65968-P “Geometría algebraica y analítica y aplicaciones”).
Communicated by: R. Cavalieri
References
[1] E. M. Bese, On the spannedness and very ampleness of certain line bundles on the blow-ups of
[2] M. Demazure, Surfaces de del Pezzo, II–V. In: Séminaire sur les Singularités des Surfaces, volume 777 of Lecture Notes in Math., 21–69, Springer 1980. MR579026 Zbl 0415.0001010.1007/BFb0085875Search in Google Scholar
[3] T. Fujita, Classification theories of polarized varieties. Cambridge Univ. Press 1990. MR1162108 Zbl 0743.1400410.1017/CBO9780511662638Search in Google Scholar
[4] R. Hartshorne, Algebraic geometry. Springer 1977. MR0463157 Zbl 0367.1400110.1007/978-1-4757-3849-0Search in Google Scholar
[5] P. Ionescu, Embedded projective varieties of small invariants. In: Algebraic geometry, Bucharest 1982 , volume 1056 of Lecture Notes in Math., 142–186, Springer 1984. MR749942 Zbl 0542.1402410.1007/BFb0071773Search in Google Scholar
[6] A. Lanteri, R. Mallavibarrena, Osculation for conic fibrations. J. Pure Appl. Algebra 220 (2016), 2852–2878. MR3471192 Zbl 1342.1401310.1016/j.jpaa.2016.01.005Search in Google Scholar
[7] A. Lanteri, R. Mallavibarrena, R. Piene, Inflectional loci of quadric fibrations. J. Algebra 441 (2015), 363–397. MR3391932 Zbl 1327.1405410.1016/j.jalgebra.2015.06.023Search in Google Scholar
[8] A. Lanteri, M. Palleschi, About the adjunction process for polarized algebraic surfaces. J. Reine Angew. Math. 352 (1984), 15–23. MR758692 Zbl 0535.1400310.1515/crll.1984.352.15Search in Google Scholar
[9] M. Nagata, On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1. Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 32 (1960), 351–370. MR126443 Zbl 0100.1670310.1215/kjm/1250776405Search in Google Scholar
[10] I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. of Math. (2) 127 (1988), 309–316. MR932299 Zbl 0663.1401010.2307/2007055Search in Google Scholar
[11] I. R. Shafarevich et al., Algebraic surfaces. Amer. Math. Soc. 1965. MR0215850 Zbl 0154.21001Search in Google Scholar
[12] K. Yokoyama, On blowing-up of polarized surfaces. J. Math. Soc. Japan 51 (1999), 523–533. MR1694783 Zbl 0968.1400310.2969/jmsj/05130523Search in Google Scholar
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