Consider the poset $P_I=\text{\rm Borel}(\Bbb R)\setminus I$ where $I$ is an arbitrary $\sigma$-ideal $\sigma$-generated by a projective collection of closed sets. Then the $P_I$ extension is given by a single real $r$ of an almost minimal degree: every real $s\in V[r]$ is Cohen-generic over $V$ or $V[s]=V[r]$.
@article{119311,
author = {Jind\v rich Zapletal},
title = {Forcing with ideals generated by closed sets},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {43},
year = {2002},
pages = {181-188},
zbl = {1069.03037},
mrnumber = {1903318},
language = {en},
url = {http://dml.mathdoc.fr/item/119311}
}
Zapletal, Jindřich. Forcing with ideals generated by closed sets. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 181-188. http://gdmltest.u-ga.fr/item/119311/
Set Theory: On the Structure of the Real Line, (1995), A K Peters Wellesley, Massachusetts. (1995) | MR 1350295 | Zbl 0834.04001
Set Theory, (1978), Academic Press New York. (1978) | MR 0506523 | Zbl 0419.03028
A proof of projective determinacy, J. Amer. Math. Soc. (1989), 85 6582-6586. (1989) | MR 0959109 | Zbl 0668.03021
Proper forcings and absoluteness in $L(\Bbb R)$, Comment. Math. Univ. Carolinae (1998), 39 281-301. (1998) | MR 1651950 | Zbl 0939.03054
Covering analytic sets by families of closed sets, J. Symbolic Logic 59 (1994), 1022-1031. (1994) | MR 1295987 | Zbl 0808.03031
Supercompact cardinals, sets of reals and weakly homogeneous trees, Proc. Natl. Acad. Sci. USA 85 6587-6591 (1988). (1988) | MR 0959110 | Zbl 0656.03037
Isolating cardinal invariants, J. Math. Logic accepted. | Zbl 1025.03046
Countable support iteration revisited, J. Math. Logic submitted.