An example of a family of projective irreducible curves I'd like to construct explicitely (namely with a parametric equation) the following example:
A family of projective curves parameterized by $\mathbb P^1(\mathbb C)$ with 3 properties:


*

*All curves are irreducible. 

*All but finitely many curves are smooth of genus $g>1$, and the singular curves have at most a (single) node.

*There is at least a section isomorphic to $\mathbb P^1(\mathbb C)$ which meets each curve transversally.


Practically I want an explicit example of an irreducible Lefschetz fibration of genus $g>1$.
Do you have any hint?
 A: Let $V$ be a 3-dimensional vector space and $L_0\subset V$, a fixed 1-dimensional subspace. Consider the canonical morphism (a $\mathbf{G}_m$-quotient) $\phi\colon  V\setminus (0)\to \mathbf{P}(V)$. Now for the family of 2-dimensional subspaces containing $L_0$, their images under $\phi$ might be the family of $\mathbf{P}^1$ you are looking for. They all contain the point $\phi(L_0)$, and they are all projective lines, hence smooth irreducible.
EDIT: Revision to handle curves of higher genus. Take some projective space $ \mathbf{P}^n$, and an embedded curve $X$ of higher genus. Fix a point $x_0\in X$. Now look at the automorphism group $PGL(n+1)$ of the projective space. 
Let $H$ be the isotropy subgroup at $x_0$.
Define $Y= \overline {\{h.x \mid h\in H, x\in X \}}$, bar denoting the Zariski closure. This should have the properties you want, it might be a much larger family. Is too large a family undesirable? 
Actually nothing about curves, or their genus, is used here. One can replace $X$ by any variety having some prescribed property, and that will be shared in the family as they are all isomorphic copies of it.
