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I've been going through a bit of the literature on the classification of Fano varieties, and it was pointed out to me by one of my committee members that I overlooked an important question regarding t …

Following the construction of [2], consider the full subcategory $\mathcal{D}_0 \subset D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^1})$ consisting of complexes whose cohomology objects are supporte …

Suppose $X$ is a prime Fano threefold of index 1 such that $H = -K_X$ is ample. There is a full classification of the derived category of such threefolds depending on the genus of $X$; in the case tha …

As explained in the comments of this answer, given a smooth Fano 3-fold of index 1 and genus $g \geq 6$, we have two semiorthogonal decompositions $$\langle \text{Ku}(X), \mathcal{E}, \mathcal{O}_X\ra …

Let $X$ be an unnodal Enriques surface together with an isotropic 10-sequence $\{ f_1, \dots, f_{10}\} \subset \operatorname{Num}(X)$, and let $F_i^\pm \in \operatorname{NS}(X)$ denote the two preimag …

I've been reading through a bit of the literature on stability conditions, and one of the models that has come up is the 'local projective plane'. Explicitly, this is the total space of the canonical …

This question is motivated by the construction of the Kuznetsov component on a prime Fano threefold $X$ of index 1 (say genus $g \geq 6$, $g \neq 7, 9$): $$ D^b(X) = \langle Ku(X), E, \mathcal{O}_X \r …

For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category : $$ S_X : D^\flat(X) \to D^\flat(X), \hspace{3em} S_X(-) = - \otimes \omega_X[\dim X] $$ sati …

I apologize in advance for the long title. This question is motivated primarily by [2], with the explicit example of $\mathbb{P}^2$ and $\omega_{\mathbb{P}^2}$ computed in [3] and [1], respectively. L …