# $$Pic(X)$$ vs. $$CP^infty$$

There are likely inaccuracies in this post, as I wrote it quickly and am just beginning to learn the basics of algebraic geometry. Constructive criticism is strongly encouraged.

As we saw in a Precursor to Characteristic Classes, $$CP^\infty$$ is the classifying space of complex line bundles over $$X$$.

$$CP^\infty$$ is, in some sense, the moduli space of line bundles over a point. There’s only one isomorphism class of line bundles over a point — but then this one line bundle has automorphism group $$C^\times$$ (which is homotopy equivalent to $$U(1)$$).

Allow me to introduce you to something that looks a LOT like $$CP^\infty$$.

What is this map, $$p \times C \to Pic(C) \times C$$, you might ask. Choose a point on our curve $$C$$ and this defines a line bundle over $$S$$ corresponding to a choice of the class of line bundles in $$Pic(C)$$. In other words, we take a point on a (not sure if I require smoothness here) algebraic curve and turn it into a line bundle on that curve.

Warning: I’ve been told that there is a difference between topological line bundles and algebraic line bundles, unfortunately, I don’t know why or what it is! I mention this, for $$Pic(X)$$ usually corresponds to *algebraic* line bundles over $$X$$.

Why is the multiplicative formal group getting involved? Let’s briefly review what the multiplicative formal group law is (as a group scheme).

Thank you to Edward Frenkel for kindly explaining the difference between $$CP^\infty$$ and $$Pic(X)$$ (both classifying spaces of line bundles), and to Qiaochu Yuan for explaining why on earth $$CP^\infty$$ is the moduli space of line bundles over a point. Any errors are mine, not theirs.