I would like to know something about the following two questions.

  1. Given $X$ Fano manifold and $L$ an ample line bundle on $X$, we define \begin{gather} \sigma(L,x):=\sup\{t>0\, :\, \mu^{*}L-tE \,\, \mbox{is big}\, \},\\ \end{gather} for any $x\in X$ where $\mu:Y\to X$ is the blow-up at $x$ and $E$ is the exceptional divisor. Is there any known upper bound on $$ \sigma(L):=\sup_{x\in X}\sigma(L,x)\, ? $$

  2. In the same setting of point one and with the same notations, letting \begin{gather} \epsilon(L,x):=\sup\{t>0:\, \mu^{*}L-tE\, \,\mbox{is ample}\,\} \end{gather} to be the one-point Seshadri constant of $L$ at $x\in X$, is there any known lower bound on $\epsilon(L):=\inf_{x\in X} \epsilon(L,x)$?

An uniform bound depending uniquely on $n:=\dim X$,$(L^n)$ and on $(-K_{X}^{n})$ would be great, but I believe it may be too optimistic. So any comment is very welcome, also regarding particular Fano manifolds.

An important remark. It is possible to prove that $\sigma(L)\epsilon(L)^{n-1}\leq (L^{n})$ so the second question is stronger than the first one.

As particular example, consider $X=S_{r}$ to be a surface given by the blow-up of $P^{2}$ at $r$-general point $\{p_{1},\dots,p_{r}\}$ for $1\leq r\leq 9$ and let $L:=3H-E_{1}-\cdots-E_{r-1}-\delta E_{r}$ for $\delta$ rational such that $dL$ is ample for $d$ natural number divisible enough. What is it known about \begin{gather} \sigma(dL),\epsilon(dL)? \end{gather} Note that $S_{r}$ is a Del Pezzo surface for $r\leq 8$, while $S_{9}$ is not a Fano manifold but I would like to known more about the quantity described above also for this manifold.

Thanks in advance!!!


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