Some interesting tiny facts (1)

This is just a collection of some tiny cute explanation to 2 small questions: visualizing Euler sequence and symmetric products.

By Euler sequence I mean the following familiar exact sequence in algebraic geometry:

where one can also look at the dualized version (when it is dualizable):

The sequence is handy and fundamental, but as Professor Ravi Vakil said in his notes in algebraic geometry, “while I can prove it, I don’t understand in my bones why it is true”. Recently when I went over the review of math preliminaries in the book of Mirror symmetry by K.Hori, S. Katz, etc., I found an interesting explanation given there, and anny enough, Ravi Vakil is also among the editor of this book…

Here’s the explanation of the book to this sequence: if we take the point of view that the homogeneous coordinate of the projective space is just (n+1) sections of , then it is actually (n+1) sections of . For such a section, we obtain a vector field (Einstein convention here) over which descends to projective space through the canonical projection. Thus we have the mapping of the last 2 terms. While checking the last step, one also finds that the kernel is exactly the constant field, which concludes the argument.

With some careful thought one also see the explanation works also for Grassmannian, and to check the vector field constructed over descends to projective space is almost identical to the calculation in Ravi Vakil’s notes.

I am expecting to put notes like this once and a while in here, not to give big theories but just interesting explanation or constructions interests me. I hope I will blog about symplectic cuts and symmetric products in the future week. Feel free to insert your notes between mine, as long as you don’t find this too trivial. 🙂