A lower bound on Seshadri constants of hyperplane bundles on threefolds (Q848824)

Given a projective variety \(X\) of dimension \(d\) and a nef line bundle \(L\) on \(X\). Let \(x\) be a point on \(X\) and \(\pi: \tilde{X}\rightarrow X\) the blowup of \(X\) at \(x\) with the exceptional divisor \(E\). The Seshadri constant \(\epsilon (L, x)=\sup\{\alpha \geq 0\mid \pi^*L-\alpha E {\text{ is nef}}\}\). It measures how positive a nef line bundle locally is near a given point. It is also interesting on its own. In this paper, the author proves a similar result to Bauer's result for a surface and gives the lower bound on Seshadri constants for very ample line bundles on smooth projective threefolds. A threefold of degree \(d\) in \({\mathbb{P}}^4\) with finitely many singular points is constructed such that its Seshadri constant of a hyperplane bundle at a smooth point is \(\frac{d}{d-1}\).