On the maximal variation problem and Lefschetz pencils

Davide Bricalli; Gian Pietro Pirola

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arXiv · 2606.27893

On the maximal variation problem and Lefschetz pencils

Davide Bricalli·Gian Pietro Pirola

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Abstract

We study the maximal variation problem for linear systems associated with a very ample line bundle, using Hodge theory and Picard-Lefschetz theory. We provide an affirmative answer to the maximal variation problem for a broad class of smooth projective varieties. This includes varieties $X$ of dimension $n\geq2$ with $p_g=h^{n,0}(X)>0$ and $H^{n-1,0}(X)=\{0\}$, Enriques surfaces, irregular surfaces with maximal Albanese dimension, smooth hyperk\"ahler varieties, and all the smooth not Fano hypersurfaces in $\mathbb{P}^n$. As a consequence, by a result of Beauville, we establish a Lefschetz property for the Jacobian rings of smooth hypersurfaces in $\mathbb{P}^n$ of degree n+1.

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Davide Bricalli, Gian Pietro Pirola. 2026-06-26. On the maximal variation problem and Lefschetz pencils. https://arxiv.org/abs/2606.27893

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