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Let me start with concrete examples. Let $X$ be a smooth special Gushel-Mukai threefold and $\mathcal{C}(X)$ be its honest Fano surface of conics, it has two irreducible components $\mathcal{C}(X)=\...

The motivation for this question is that I would like to extract some information from derived category of surfaces to conclude the rationality of surface. There is a well known rationality criterion ...

Let $X$ be a smooth projective $3$-fold over $\mathbb{C}$. Let $K_0(X)$ be its Grothendieck group, consider the Euler form defined as: $\chi(M,N): K_0(X)\times K_0(X)\rightarrow\mathbb{Z}$ by $(M,N)\...