[Paper Review] Strictly nef divisors and some remarks on a conjecture of Serrano

Priyankur Chaudhuri|arXiv (Cornell University)|Aug 11, 2020

Algebraic Geometry and Number Theory13 references4 citations

TL;DR

This paper establishes generalized abundance for smooth Fano fibrations over varieties of general type and proves a generalized version of Serrano's conjecture for surfaces with almost strictly nef line bundles. It shows that under the SHGH conjecture, strictly nef non-ample divisors exist on rational surfaces, providing new examples and advancing the understanding of strictly nef divisors in algebraic geometry.

ABSTRACT

Serrrano's Conjecture says that if $L$ is a strictly nef line bundle on a smooth projective variety $X$, then $K_X+tL$ is ample for $ t > dim X+1$. In this paper I will prove a few cases of this conjecture. I will also prove a generalized version of this conjecture (due to Campana, Chen and Peternell) for surfaces. In the last section, assuming the SHGH conjecture, I will give a series of examples of strictly nef non ample divisors on surfaces of arbitary Kodaira dimension.

Motivation & Objective

• To prove generalized abundance for smooth Fano fibrations over varieties of general type, extending known results on strictly nef divisors.
• To verify a generalized form of Serrano's conjecture for surfaces with almost strictly nef line bundles.
• To demonstrate that the SHGH conjecture implies the existence of strictly nef non-ample divisors on rational surfaces.
• To investigate the ampleness and bigness of adjoint divisors $K_X + tL$ for strictly nef $L$ in various geometric settings.
• To provide new examples of strictly nef non-ample divisors via blow-ups of $\mathbb{P}^2$ under the SHGH conjecture.

Proposed method

• Uses the Cone Theorem to show that $K_X + tL$ is strictly nef for $t > d+1$ when $X$ is smooth and $L$ is strictly nef.
• Applies the numerical criterion from Lemma 7 to detect failure of semi-ampleness when $K_X^d = \cdots = L^d = 0$.
• Employs resolution of indeterminacy and Iitaka fibrations to analyze the bigness of $K_X + tL$ via pullbacks to smooth models.
• Applies the SHGH conjecture to compute expected dimensions of linear systems on blow-ups of $\mathbb{P}^2$, ensuring positivity of intersection numbers.
• Constructs strictly nef non-ample divisors via base change of finite morphisms from $S$ to $\mathbb{P}^2$, pulling back a strictly nef divisor on the blow-up.
• Uses the fact that $L \cdot C > 0$ for all curves $C$ in the sphere defined by the SHGH conjecture to verify strict nefness.

Experimental results

Research questions

• RQ1 Does generalized abundance hold for smooth Fano fibrations over varieties of general type?
• RQ2 Is $K_S + tL$ big for all $t > 3$ when $L$ is almost strictly nef on a surface $S$?
• RQ3 Does the SHGH conjecture imply the existence of strictly nef non-ample divisors on rational surfaces?
• RQ4 Under what conditions is a strictly nef divisor $L$ on a surface necessarily big?
• RQ5 Can the adjoint divisor $K_X + tL$ be shown to be big for $t \gg 0$ when $\kappa(aK_X + bL) \geq \dim X - 2$?

Key findings

• Generalized abundance holds for smooth Fano fibrations over varieties of general type: if $-K_X$ is ample on fibers and $Y$ is of general type, then $K_X + L$ is semiample when $L$ and $K_X + L$ are nef.
• For a smooth projective surface $S$ with an almost strictly nef line bundle $L$, $K_S + tL$ is big for all $t > 3$, and this extends to surfaces with Gorenstein singularities.
• Under the SHGH conjecture, for any projective surface $S$, there exists a surface $S'$ birational to $S$ admitting a strictly nef non-ample divisor.
• If $\kappa(L) \geq \dim X - 2$, then $L$ is big, providing a numerical criterion for bigness of strictly nef divisors.
• For any smooth projective variety $X$ with $\kappa(aK_X + bL) \geq \dim X - 2$, the divisor $K_X + tL$ is big for all $t \gg 0$.
• The construction of strictly nef non-ample divisors relies on the SHGH conjecture to ensure that the expected dimension of linear systems on blow-ups of $\mathbb{P}^2$ is non-negative, guaranteeing positivity of intersections with curves.