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Effective adjunction theory

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Abstract

Here we investigate the property of effectivity for adjoint divisors. Among others, we prove the following results: A projective variety X with at most canonical singularities is uniruled if and only if for each very ample Cartier divisor H on X we have \(H^0(X, m_0K_X+H)=0\) for some \(m_0=m_0(H)>0\). Let X be a projective 4-fold, L an ample divisor and t an integer with \(t \ge 3\). If \(K_X+tL\) is pseudo-effective, then \(H^0(X, K_X+tL) \ne 0\).

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Author notes

  1. Claudio Fontanari

    Present address: Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38123, Trento, Italy

Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38123, Trento, Italy

    Marco Andreatta

Authors
  1. Marco Andreatta
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  2. Claudio Fontanari
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Corresponding author

Correspondence to Claudio Fontanari.

Additional information

We would like to thank Paolo Cascini, Roberto Pignatelli and Luis Sola-Conde for fruitful conversations. We are grateful to János Kollár for pointing out his examples and for suggesting projective varieties with canonical singularities as a good category to settle our results. We also thank the referees for useful comments. The research project was partially supported by GNSAGA of INdAM, by PRIN 2015 “Geometria delle varietà algebriche”, and by FIRB 2012 “Moduli spaces and Applications”.

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Andreatta, M., Fontanari, C. Effective adjunction theory. Ann Univ Ferrara 64, 243–257 (2018). https://doi.org/10.1007/s11565-018-0300-z

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