Related papers: Pentagramma Mirificum. II
In this paper, we show that there is a natural Lagrangian fibration structure on the map $\Phi$ from the cotangent bundle of a del Pezzo surface $X$ of degree 4 to $\mathbb C^2$. Moreover, we describe explicitly all level surfaces of the…
The classification of minimal rational surfaces and the birational links between them by Iskovskikh, Manin and others is a well-known subject in the theory of algebraic surfaces. We explain algorithms that realise links of type II between…
The Fano models of Enriques surfaces produce a family of tens of mutually intersecting planes in $\mathbf P^5$ with a $10$-dimensional moduli space. The latter is linked to several 10-dimensional moduli spaces parametrizing other types of…
We compute the CM volume, that is the degree of the descended CM line bundle, of the Fano K-moduli space of Quartic del Pezzo in any dimension, and of the K-moduli space of the log Fano hyperplane arrangements of dimension one and two.…
We complete the classification of automorphism groups of del Pezzo surfaces over algebraically closed fields of odd positive characteristic.
Over the projective plane and at most two-step blowups of Hirzebruch surfaces, where there are strong full exceptional sequences of line bundles, we obtain foundational results about Gaeta resolutions of coherent sheaves by these line…
Mysterious duality is a relationship, described by Vafa in 2000, between $\frac12$-BPS branes in Type II supergravity in dimension $D=d+2$ and rational curves on del Pezzo surfaces of degree $d$. We show that both sides of this…
This is an informal set of lecture notes on moduli spaces of curves based on a set of lectures given at the ICTP last summer. It begins at an elementary level and discusses the genus 1 case in detail. The notes then give an informal…
The present work introduces an efficient Monte Carlo algorithm for continuum percolation composed of randomly-oriented rectangles. By conducting extensive simulations, we report high precision percolation thresholds for a variety of…
In this paper, we shall find a new connection between $n$th degree polynomial mod $p$ congruence with $n$ roots and higher-order Fibonacci and Lucas sequences. We shall first discuss the recent work been done in sequences and their…
More strong version of the main inductive theorem about the complements on surfaces is proved and the models of exceptional log del Pezzo surfaces with $\delta=0$ are constructed
We use hypersurfaces containing unexpected linear spaces to construct interesting vector bundles on complete intersection surfaces in projective space. We discover examples of moduli spaces of rank 2 stable bundles on surfaces of Picard…
These are lecture notes based on a series of talks given by the authors at the CIMPA Summer School on Algebraic Geometry and Hypergeometric Functions held in Istanbul in Summer of 2005. They provide an introduction to a recent work on the…
On del Pezzo surfaces, we study effective ample $\mathbb{R}$-divisors such that the complements of their supports are isomorphic to $\mathbb{A}^1$-bundles over smooth affine curves.
Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for…
We classify geometrically integral regular del Pezzo surfaces which are not geometrically normal over imperfect fields of positive characteristic. Based on this classification, we show that a three-dimensional terminal del Pezzo fibration…
We classify smooth del Pezzo surfaces whose alpha-invariant of Tian is bigger than one.
In this paper I will explain a rigidity conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. I will also establish a simple case of the conjecture, the one involving the $3$-diagonal map on a convex $8$-gon…
We show that smooth quintic del Pezzo threefolds over arbitrary base schemes are classified by non-degenerate ternary symmetric bilinear forms. Then we describe the automorphism group schemes, the Hilbert schemes of lines and the orbit…
We formulate three versions of a strange duality conjecture for sections of the Theta bundles on the moduli spaces of sheaves on abelian surfaces. As supporting evidence, we check the equality of dimensions on dual moduli spaces, answering…