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This may be a very basic question, so my apologies if that is the case.

But I was interested in having some examples of meromorphic (singular) maps into complex projective space from complex surfaces (or higher dimensional objects) which are in the Sobolev space $W^{1,2}$.

The prototype of the examples I want would be the rational map $(z_1,z_2) \mapsto [z_1:z_2]$ from $\mathbb{C}^2 \rightarrow \mathbb{C} P^1$. This is well-defined and holomorphic away from the origin, cannot be extended to the origin and has the desired regularity.

I hoped that I could also build a map by using some explicit examples of globally generated holomorphic line bundles on complex surfaces. For this I would want to take a subcollection of the generating set of sections, say $s_i$, which vanish at some point $p_0 \in M$ and consider the map $p \mapsto [s_1(p):s_2(p):...:s_m(p)]$. The regularity of this should just be governed by the regularity of the sections near the vanishing point.

Any help would be appreciated, or some specific cases to look at would be great.For example it would be useful to know as many (explicit?) examples of ample line bundles on surfaces with $h^0 \geq 3$.

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  • $\begingroup$ The map $[z_1 : z_2 : z_3] \mapsto [z_1 : z_2]$ that you mention is a rational map $CP^2 \dashrightarrow CP^1$ with locus of indeterminacy $\{(0, 0)\} \in CP^2$ -- there's no way to define it at (0,0), since the map can take on any ratio as $z_1$ and $z_2$ approach 0. $\endgroup$ – Kevin Casto Apr 15 at 18:46
  • $\begingroup$ Yes you're right it's a rational map but this is fine. But what I say in the second paragraph is mostly nonsense so I have edited. $\endgroup$ – ben Apr 16 at 14:29

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