According to Katz & Mazur's book "Arithmetic Moduli of Elliptic Curves", an elliptic curve $E$ over an arbitrary scheme $S$ is a proper smooth curve over $S$ with geometrically connected fibers all of genus $1$ and given with a section $"0"$. Note that "smooth curve" is defined as a smooth morphism $E\rightarrow S$ of relative dimension $1$ which is separated and of finite presentation.
In Silverman's book "The Arithmetic of Elliptic Curves" however, an elliptic curve is a pair $(E,O)$ where $E$ is a smooth curve of genus $1$ and $O\in E$. By "curve", it is meant a projective variety of dimension $1$.
It seems to me that both definition differ concerning the hypothesis of being geometrically connected. I know that an elliptic curve over a field (eg. $\mathbb{R}$) may not be connected, but what about an algebraically closed field? Using Silverman's definition, is it true that an elliptic curve may not be geometrically connected?
Thank you very much for you help.