# On the set of divisors with zero geometric defect

## TL;DR

This paper proves that for a transcendental holomorphic curve into a complex projective manifold, the zero divisor of a generic section of a very ample line bundle has zero geometric defect, and the curve almost misses certain subsets.

## Contribution

It establishes the zero geometric defect property for divisors associated with generic sections and shows the curve's near-miss behavior for certain analytic subsets.

## Key findings

- Zero geometric defect of divisors with respect to the curve
- Curve almost misses general codimension 2 subsets
- Results hold for generic sections of very ample line bundles

## Abstract

Let $f: \mathbb{C} \to X$ be a transcendental holomorphic curve into a complex projective manifold $X$. Let $L$ be a very ample line bundle on $X$. Let $s$ be a very generic holomorphic section of $L$ and $D$ the zero divisor given by $s$. We prove that the \emph{geometric} defect of $D$ (defect of truncation $1$) with respect to $f$ is zero. We also prove that $f$ almost misses general enough analytic subsets on $X$ of codimension $2$.