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When we say, that, say, a surface contains ${\infty}^{k}$ lines, do we mean that it contains a k-parameter family of lines? Do we assume that this family is parametrized by a $P^{k}$, say, or we use this term more informally? This is certainly a standard notation, but I didn't see its explanation in standard modern textbooks.

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I have never seen the notation $\infty^k$ in algebraic geometry. Could you provide a reference? –  Pete L. Clark Mar 16 '10 at 14:26
@Pete: Arbarello, Cornalba, Griffiths and Harris, page 13. –  Alberto García-Raboso Mar 16 '10 at 14:54

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up vote 3 down vote accepted

The terminology is fairly common in classical works on projective algebraic/differential geometry. I am not aware of its origins. Anyway it is used rather informally and only means that you have a $k$-dimensional "continuous" family of objects. The parameter space usually is only local and should be thought as a small ball in $\mathbb C^k$.

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Thank you, Jorge. –  rat Mar 16 '10 at 18:40
You are welcome, rat. –  jvp Mar 16 '10 at 19:58

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