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Normal canonical hypersurfaces

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Abstract

We construct normal, canonical hypersurfaces \(V \subset \mathbbm {P}^d\), for any \(d \ge 4\), of degree \(d+3\), giving the explicit equation for V. The existence of the above hypersurfaces was not known. The first plurigenera and the irregularities (that are all vanishing) of a desingularization \(X \rightarrow V\) of V are calculated. These hypersurfaces are birationally distinct from the trivial case of hypersurfaces of degree \(d+2\) that are nonsingular, or endowed with negligible singularities. A concise update on the classification of canonical surfaces in \(\mathbbm {P}^3\) is given in the Introduction, and a rather heavy omission is pointed out.

Sunto

Si costruiscono ipersuperficie canoniche, normali \(V \subset \mathbbm {P}^d\), per ogni \(d \ge 4\), di ordine \(d+3\), la cui esistenza non era nota, fornendo l’equazione esplicita di V. Si calcolano i primi plurigeneri e le irregolarità, che risultano tutte nulle, di una desingolarizzazione \(X \rightarrow V\) di V. Tali ipersuperficie sono birazionalmente distinte dal caso banale di ipersuperficie di grado \(d+2\) non singolari o dotate di singolarità trascurabili. Nell’Introduzione diamo un breve aggiornamento sulla classificazione delle superficie canoniche di \(\mathbbm {P}^3\), facendo notare una omissione piuttosto grave.

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Acknowledgments

The author is deeply grateful to Professor Kazuhiro Konno for informing him about Zariski’s example and his own studies on normal, canonical surfaces in \(\mathbbm {P}^3\), introducing the author to research on normal, canonical hypersurfaces in \(\mathbbm {P}^d\), \(d \ge 4\). In fact, the hypothesis of normality enables the theory of pluricanonical adjoints to the hypersurfaces to be used to compute the plurigenera of their desingularizations, and this theory is quite familiar to the author. Said theory is revisited and developed in [35].

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  1. Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Università di Padova, Stradella S. Nicola, 3, 36100, Vicenza, Italy

    Ezio Stagnaro

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  1. Ezio Stagnaro
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Correspondence to Ezio Stagnaro.

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The present work was the object of a talk at the Kulikov Conference at the Steklov Institute in Moscow, December 3–7, 2012.

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Stagnaro, E. Normal canonical hypersurfaces. Rend. Circ. Mat. Palermo, II. Ser 65, 365–385 (2016). https://doi.org/10.1007/s12215-016-0239-9

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