Kuznetsov’s Fano threefold conjecture via Hochschild–Serre algebra

Let Y be a smooth quartic double solid regarded as a degree 4 hypersurface of the weighted projective space \(\mathbb {P}(1,1,1,1,2)\) $\mathbb{P}(1,1,1,1,2)$ . We study the multiplication of the Hochschild-Serre algebra of its Kuznetsov component \(\mathcal {K}u(Y)\) $\mathcal{K}u(Y)$ via matrix factorization. As an application, we give a new disproof of Kuznetsov’s Fano threefold conjecture.