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Register a new userBlowing up linear categories, refinements, and homological projective duality with base locus
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In this paper, we first introduce geometric operations for linear categories, and as a consequence generalize Orlovs blow up formula [O04] to possibly singular local complete intersection centres. Second, we introduce refined blowing up of linear category along base--locus, and show that this operation is dual to taking linear section. Finally, as an application we produce examples of Calabi--Yau manifolds which admits Calabi--Yau categories fibrations over projective spaces.
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Homological Projective duality (HP-duality) theory, introduced by Kuznetsov [42], is one of the most powerful frameworks in the homological study of algebraic geometry. The main result (HP-duality theorem) of the theory gives complete descriptions of bounded derived categories of coherent sheaves of (dual) linear sections of HP-dual varieties. We show the theorem also holds for more general intersections beyond linear sections. More explicitly, for a given HP-dual pair $(X,Y)$, then analogue of HP-duality theorem holds for their intersections with another HP-dual pair $(S,T)$, provided that they intersect properly. We also prove a relative version of our main result. Taking $(S,T)$ to be dual linear subspaces (resp. subbundles), our method provides a more direct proof of the original (relative) HP-duality theorem.
We investigate the blow-up of a weighted projective plane at a general point. We provide criteria and algorithms for testing if the result is a Mori dream surface and we compute the Cox ring in several cases. Moreover applications to the study of $overline{M}_{0,n}$ are discussed.
Let $Z$ be a closed subscheme of a smooth complex projective variety $Ysubseteq Ps^N$, with $dim,Y=2r+1geq 3$. We describe the intermediate Neron-Severi group (i.e. the image of the cycle map $A_r(X)to H_{2r}(X;mathbb{Z})$) of a general smooth hypersurface $Xsubset Y$ of sufficiently large degree containing $Z$.
Given a gauged linear sigma model (GLSM) $mathcal{T}_{X}$ realizing a projective variety $X$ in one of its phases, i.e. its quantum Kahler moduli has a maximally unipotent point, we propose an emph{extended} GLSM $mathcal{T}_{mathcal{X}}$ realizing the homological projective dual category $mathcal{C}$ to $D^{b}Coh(X)$ as the category of B-branes of the Higgs branch of one of its phases. In most of the cases, the models $mathcal{T}_{X}$ and $mathcal{T}_{mathcal{X}}$ are anomalous and the analysis of their Coulomb and mixed Coulomb-Higgs branches gives information on the semiorthogonal/Lefschetz decompositions of $mathcal{C}$ and $D^{b}Coh(X)$. We also study the models $mathcal{T}_{X_{L}}$ and $mathcal{T}_{mathcal{X}_{L}}$ that correspond to homological projective duality of linear sections $X_{L}$ of $X$. This explains why, in many cases, two phases of a GLSM are related by homological projective duality. We study mostly abelian examples: linear and Veronese embeddings of $mathbb{P}^{n}$ and Fano complete intersections in $mathbb{P}^{n}$. In such cases, we are able to reproduce known results as well as produce some new conjectures. In addition, we comment on the construction of the HPD to a nonabelian GLSM for the Plucker embedding of the Grassmannian $G(k,N)$.
We prove that the generic point of a Hilbert modular four-fold is not a Jacobian. The proof uses degeneration techniques and is independent of properties of the mapping class group used in preceding papers on locally symmetric subvarieties of the moduli space of abelian varieties contained in the Schottky locus.
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