In Cox and Katz's mirror symmetry and algebraic geometry, it is claimed that the equation of a quintic threefold $V$ in $\mathbb P^4$ gives a section of rank 6 vector bundle $\operatorname{Sym}^5(U^*)$ over the Grassmannian of lines in $\mathbb P^4$, where $U$ is the tautological bundle on the Grassmannian. This looks similar to how sections of $\mathcal O_{\mathbb P^n}(5)$ are given by homogeneous polynomials, but I am unsure of how to see this (for Grassmannian). It seems like it is possible to extract information on Grassmannian using Borel-Weil-Bott, according to https://mathoverflow.net/questions/240378/global-section-of-universal-bundle-on-grassmanian
Is there a straightforward description like the case for $\mathbb P^n$? I thought of the following:
$\require{AMScd}$
\begin{CD}
\operatorname{Sym}^5(U^*) @>?>> ???\\
@VVV @VVV\\
\operatorname{Gr}(2,5)\backslash S @>>> \operatorname{Sym}^5V
\end{CD}
On a general quintic threefold, there are 2875 lines, corresponding to the set of points $S$, outside those points, there are 5 intersection $\ell\cap V\in \operatorname{Sym}^5 V$. If we can realize $\operatorname{Sym}^5(U^*)$ as pullback on some natural bundle on $\operatorname{Sym}^5V$, then maybe it is possible to write down a natural section on $\operatorname{Sym}^5(U^*)$. But I know almost nothing about $\operatorname{Sym}^5 V$, does anyone have an idea? Thank you.
Best Answer
There are two ways of constructing a section of $\operatorname{Sym}^5 U^*$. The first is to realize the degree 5 polynomial defining $V$ as an element $\operatorname{Sym}^5(\mathbb C^5)^*$. Therefore from Euler sequence on Grassmannian, $$0\to U\to G(2,5)\times \mathbb C^5\to Q\to 0$$
Taking dual and symmetric power, we get $$\operatorname{Sym}^5(\mathbb C^5)^*\to\operatorname{Sym}^5U^*$$
And we can think of the polynomial as a constant section of the left.
Another construction is to identify the fiber of $\operatorname{Sym}^5U^*$ over $[\ell]\in G(2,5)$ as $H^0(\ell, i^*(\mathcal O_{\mathbb P^4}(5)))=H^0(\ell, \mathcal O_{\ell}(5))$. Here $\ell$ is a line in $\mathbb P^4$, and $i$ is the inclusion.