On the minimal number of singular fibers with non-compact Jacobians for families of curves over $mathbb P^1$

Let $f:X to mathbb{P}^1$ be a non-isotrivial family of semi-stable curves of genus $ggeq 1$ defined over an algebraically closed field $k$ with $s_{nc}$ singular fibers whose Jacobians are non-compact. We prove that $s_{nc}geq 5$ if $k=mathbb C$ and $ggeq 5$; we also prove that $s_{nc}geq 4$ if ${rm char}~k>0$ and the relative Jacobian of $f$ is non-smooth.

Stability of the LCD Model

In this paper, first-passage probability of Markov chains is used to get a strict proof of the existence of degree distribution of the LCD model presented by Bollobas (Random Structures and Algorithms 18(2001)). Also, a precise expression of degree distribution is presented.