# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 19th 2021
• (edited Jul 19th 2021)

starting something, on the kind of theorems originating with

Nothing to be seen here yet, but I need to save. (Am not sold on the entry title, except that “topology” is not really the right term here.)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 19th 2021

added statement (here) of Segal’s original theorem from 1979.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 19th 2021

added a remark (here) highlighting the similarity/difference to the statement of the homotopical Oka principle for non-compact domain surfaces.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 19th 2021
• (edited Jul 19th 2021)

added a brief remark cross-linking with twistor string theory (here) – to be expanded

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 28th 2021

added statement of the analogous theorem (here) for maps between projective spaces, from

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJul 29th 2021

I have added the qualifier “regular” to “rational map”, but still need to say this better – the terminology here is confusing.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJul 29th 2021

I have added more words to the Idea-section and a remark on the regularity of the rational functions.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 29th 2021
• (edited Jul 29th 2021)

I have added a remark (here) that the space of rational maps $\mathbb{C}P^1 \to \mathbb{C}P^n$ that is considered in Segal’s theorem is also considered in Gromov-Witten theory (after compactification and quotienting), as is nicely explicit in Bertram 02, p. 9.

This confluence looks like it ought to have drawn attention, but I don’t find literature in this direction.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeAug 1st 2021

re #8:

I see that it is this connection which the preprint

• David Ayala, Homological Stability among Moduli Spaces of Holomorphic Curves in Complex Projective Space (arXiv:0811.2274)

was after, before the author discovered the mistake highlighted in v2.

Mistake or not, that’s the right question to ask. But it looks like it wasn’t followed up.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeAug 15th 2021

• Jacob Mostovoy, Spaces of Rational Loops on a Real Projective Space, Transactions of the American Mathematical Society, Vol. 353, No. 5 (May, 2001), pp. 1959-1970 (jstor:221802)
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeAug 15th 2021
• (edited Aug 15th 2021)

added (here) brief statement of the main theorem from

identifying the full homotopy type of the space of pointed rational maps $\mathbb{C}P^1 \to \mathbb{C}P^n$ with that of a configuration space of points. (Am making a similar addition now to this latter entry, too.)

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeAug 16th 2021

Fixed the statement of Segal’s theorem (here), now pointing to n-equivalence (following Segal’s note on terminology on p. 44)