Related papers: Geometric helices on del Pezzo surfaces from tilti…

We show that every split del Pezzo surface of degree d=5,4,3 or 2 has a universal torsor which is a dense open subset of the intersection of 6-d dilatations of the affine cone over the corresponding generalized Grassmannian G/P. Here a…

We determine which singular del Pezzo surfaces are equivariant compactifications of G_a^2, to assist with proofs of Manin's conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an…

We prove that the spaces of rational curves on del Pezzo surfaces are either irreducible or empty, with a unique exception.

We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler…

Using techniques from Bridgeland stability, we show the kernel sheaf associated to sufficiently positive Gieseker stable sheaf on a Del Pezzo surface is slope stable. This is the first effective stability result for kernel sheaves…

Based on mirror symmetry, we discuss geometric engineering of N=1 ADE quiver models from F-theory compactifications on elliptic K3 surfaces fibered over certain four-dimensional base spaces. The latter are constructed as intersecting…

We show how to construct tilting bundles for a class of smooth projective varieties using characteristic $p$ methods. Given such a variety $X$, reduce it modulo a prime number and consider the direct image of the structure sheaf under the…

Let $G$ be a finite group acting on an ice quiver with potential $(Q, F, W)$. We construct the corresponding $G$-equivariant relative cluster category and $G$-equivariant Higgs category, extending the work of Demonet. Using the orbit…

We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show the existence of purely inseparable…

We give a new proof the arithmetic Hilbert-Samuel theorem by using classical reductions in the theory of coherent sheaves, a direct proof in the case of the projective space and the conservation of some numerical invariants, called…

Let $G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p \ge 0$, and let $\mathcal{N}$ be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent $G$-orbit $C$ and…

We consider the suspension operation on Lefschetz fibrations, which takes p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant, and changes the category of the fibre (or more precisely, the subcategory consisting of…

We consider F-theory compactifications on a mirror pair of elliptic Calabi-Yau threefolds. This yields two different six-dimensional theories, each of them being nonperturbatively equivalent to some compactification of heterotic strings on…

We provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the…

We describe tilting modules of the deformed category O over a semisimple Lie algebra as certain sheaves on a moment graph associated to the corresponding block of category O. We prove that they map to Braden-MacPherson sheaves constructed…

We prove the equisingular rigidity of the singular Hirzebruch-Kummer coverings $X(n, \mathcal{L})$ of the projective plane branched on line configurations $\mathcal{L}$, satisfying some technical condition. In the case, $\mathcal{L}$ = the…

In this paper, we extend our previous work to construct (0,2) Toda-like mirrors to A/2-twisted theories on more general spaces, as part of a program of understanding (0,2) mirror symmetry. Specifically, we propose (0,2) mirrors to GLSMs on…

We study the infinitesimal rigidity of equivariant minimal maps from the universal cover of a smooth oriented surface (possibly non-compact) into a Riemannian symmetric space, focusing on representations arising from cyclic harmonic…

From a hermitian metric on the anticanonical bundle on a Del Pezzo surface, and a holomorphic section of it, we construct a one parameter family of bihermitian metrics (or equivalently generalized Kaehler structures). The construction…

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived…