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A note on gonality of curves on general hypersurfaces

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Abstract

This short paper concerns the existence of curves with low gonality on smooth hypersurfaces \(X\subset \mathbb {P}^{n+1}\). After reviewing a series of results on this topic, we report on a recent progress we achieved as a product of the Workshop Birational geometry of surfaces, held at University of Rome “Tor Vergata” on January 11th–15th, 2016. In particular, we obtained that if \(X\subset \mathbb {P}^{n+1}\) is a very general hypersurface of degree \(d\geqslant 2n+2\), the least gonality of a curve \(C\subset X\) passing through a general point of X is \(\mathrm {gon}(C)=d-\left\lfloor \frac{\sqrt{16n+1}-1}{2}\right\rfloor \), apart from some exceptions we list.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Bari, Via Edoardo Orabona 4, 70125, Bari, Italy

    Francesco Bastianelli

  2. Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Viale della Ricerca Scientifica 1, 00133, Roma, Italy

    Ciro Ciliberto & Flaminio Flamini

  3. Dipartimento di Matematica e Fisica, Università degli Studi “Roma Tre”, Largo S. L. Murialdo 1, 00146, Roma, Italy

    Paola Supino

Authors
  1. Francesco Bastianelli
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  2. Ciro Ciliberto
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  3. Flaminio Flamini
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  4. Paola Supino
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Corresponding author

Correspondence to Flaminio Flamini.

Additional information

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 307119, MIUR FIRB 2012 “Spazi di moduli e applicazioni”, MIUR PRIN 2010–2011 “Geometria delle varietà algebriche”, and INdAM (GNSAGA).

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Bastianelli, F., Ciliberto, C., Flamini, F. et al. A note on gonality of curves on general hypersurfaces. Boll Unione Mat Ital 11, 31–38 (2018). https://doi.org/10.1007/s40574-017-0129-x

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