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Register a new userSemiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields
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We study the birational properties of geometrically rational surfaces from a derived categorical point of view. In particular, we give a criterion for the rationality of a del Pezzo surface over an arbitrary field, namely, that its derived category decomposes into zero-dimensional components. For del Pezzo surfaces of degree at least 5, we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of the surface from Brauer classes and Chern classes associated to these vector bundles.
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Let S be a split family of del Pezzo surfaces over a discrete valuation ring such that the general fiber is smooth and the special fiber has ADE-singularities. Let G be the reductive group given by the root system of these singularities. We construct a G-torsor over S whose restriction to the generic fiber is the extension of structure group of the universal torsor. This extends a construction of Friedman and Morgan for individual singular del Pezzo surfaces. In case of very good residue characteristic, this torsor is unique and infinitesimally rigid.
Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of $S_L$ in Pic $S_L$, or a form of it containing the Neron-Severi torus. Let $mathcal{G}$ be the G-torsor over $S_L$ obtained by extension of structure group from a universal torsor $mathcal{T}$ over $S_L$. We prove that $mathcal{G}$ does not descend to S unless $mathcal{T}$ does. This is in contrast to a result of Friedman and Morgan that such $mathcal{G}$ always descend to singular del Pezzo surfaces over $mathbb{C}$ from their desingularizations.
We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin. We also include some explicit calculations for the projective plane, which confirm some folklore conjecture in this case.
We study compactifications of Drinfeld half-spaces over a finite field. In particular, we construct a purely inseparable endomorphism of Drinfelds half-space $Omega (V)$ over a finite field $k$ that does not extend to an endomorphism of the projective space $P (V)$. This should be compared with theorem of Remy, Thuillier and Werner that every $k$-automorphism of $Omega (V)$ extends to a $k$-automorphism of $P (V)$. Our construction uses an inseparable analogue of the Cremona transformation. We also study foliations on Drinfelds half-spaces. This leads to various examples of interesting varieties in positive characteristic. In particular, we show a new example of a non-liftable projective Calabi-Yau threefold in characteristic $2$ and we show examples of rational surfaces with klt singularities, whose cotangent bundle contains an ample line bundle.
We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces.
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