Resolution of the diagonal for projective hypersurface

Question

A famous theorem of Beilinson gives a finite, locally free resolution of the diagonal for $\mathbf{CP}^{n}$ by exterior tensor products of locally free sheaves on $\mathbf{CP}^{n}$: for $1 \leq k \leq n$, the $k$th component of the resolution is given by $\mathcal{O}(-k) \boxtimes \Omega^{k}(k)$ where $\Omega^{k}(k) := \bigwedge^{k} \Omega^{1} \otimes \mathcal{O}(k)$ (e.g. see https://johncalab.github.io/stuff/beilinson.pdf).

For a projective hypersurface $Y$ (or more generally, for a projective scheme), when does there exist a finite resolution of the diagonal in $Y \times Y$ by (direct sums of) exterior tensor products of locally free sheaves on $Y$? Does such a resolution exist, for example, in the case $Y = V(y^{2}z - x^{3} + xz^{2}) \subset \mathbf{CP}^{2}$, and if so can one explicitly write it down?

Answer 1

If there is a resolution of the diagonal of a smooth projective variety $Y$ with terms direct sums of $F'_i \boxtimes F''_i$, then it is easy to see that the Grothendieck group $K_0(Y)$ is generated by the classes of $[F'_i]$ (or of $F''_i$). To see this just consider the Fourier--Mukai transform given by the structure sheaf of the diagonal. On the one hand, it is the identity functor. On the other hand, any object of the derived category in its image has a resolution by $F'_i$. This means that any object has such a resolution, hence its class is a linear combination of $[F'_i]$.

This observation rules out any variety $Y$ whose $K_0(Y)$ is not of finite rank. For instance, it rules out any curve of positive genus, because for a curve $$ K_0(Y) = \mathbb{Z} \oplus \mathrm{Pic}(Y), $$ and if $\mathrm{Pic}^0(Y) \ne 0$, it is not of finite rank.