Alessandro Passantino | Search arXiv

arXiv2025

Effective non-vanishing for weighted complete intersections of low codimension

We show that on quasi-smooth weighted complete intersections of codimension at most 3, any ample Cartier divisor $H$ such that $H-K_X$ is ample admits a nontrivial global section. This is done by proving a generalisation of a numerical conjecture formulated by Pizzato, Sano and Tasin, which relates the existence of global sections of $H$ to the Frobenius number of the numerical semigroup generated by the weights of the ambient projective space.

math.AG↗

arXiv2024

Numerical conditions for the boundedness of foliated surfaces

We show that the set of Hilbert functions $P(m)=χ(mK_\mathcal{F})$ of 2-dimensional foliated canonical models with fixed $K_\mathcal{F}^2$, $K_\mathcal{F} \cdot K_X$ and $i_\mathbb{Q}(\mathcal{F})$ is finite. As a consequence, we deduce that two results on the effective birationality and boundedness of foliated canonical models with fixed Hilbert function still hold when only $K_\mathcal{F}^2$, $K_\mathcal{F} \cdot K_X$ and $i_\mathbb{Q}(\mathcal{F})$ are fixed. We then give examples further investigating the properties of families of canonical models, and study particular cases in which some of the conditions are not necessary.

math.AG↗