Related papers: Pentagramma Mirificum. II

Let $M_{\beta}$ denote the moduli space of stable one-dimensional sheaves on a del Pezzo surface $S$, supported on curves of class $\beta$ with Euler characteristic one. We show that the divisibility property of the Poincar\'e polynomial of…

We characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type $\mathbf{A}_3$ over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these…

In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type…

In this article we study Ceva's theorem and its higher-dimensional extensions from the perspective of algebraic and projective geometry. First, we situate the theorem within the study of algebraic surfaces by relating it to the defining…

We prove orientation results for evaluation maps of moduli spaces of rational stable maps to del Pezzo surfaces over a field, both in characteristic $0$ and in positive characteristic. These results and the theory of degree developed in a…

This paper surveys recent progress towards the Manin conjecture for (singular and non-singular) del Pezzo surfaces. To illustrate some of the techniques available, an upper bound of the expected order of magnitude is established for a…

Let k be a perfect field. Recently J.-L. Colliot-Th\'el\`ene showed that two pointless quadric surfaces over k are birationally equivalent if and only if they are isomorphic. We show that this result holds for arbitrary del Pezzo surfaces…

We introduce a family of Dirichlet series associated to real quadratic number fields that generalize the ordinary Fibonacci zeta function $\sum F(n)^{-s}$, where $F(n)$ denotes the $n$th Fibonacci number. We then give three different…

Motivated by asymptotic phenomena of moduli spaces of higher rank stable sheaves on algebraic surfaces, we study the Picard number of the moduli space of one-dimensional stable sheaves supported in a sufficiently positive divisor class on a…

For a del Pezzo surface of degree $\geq 3$, we compute the oscillatory integral for its mirror Landau-Ginzburg model in the sense of Gross-Hacking-Keel [Mark Gross, Paul Hacking, and Sean Keel, "Mirror symmetry for log Calabi-Yau surfaces…

Motivated by a question from V. Arnold about self-dual curves in projective spaces, we study {\cal M}_{m,n,k}: the moduli space of m-self-dual n-gons in {\mathbb P}^k. This paper lays out an explicit construction of self-dual polygons, and…

We study irreducibility of families of degree 4 Del Pezzo surface fibrations over curves.

We describe new explicit examples of moduli spaces of Bridgeland semistable objects on surfaces, parametrizing objects whose numerical class agrees with the class of a point. This follows ideas of Tramel and Xia, using stability conditions…

We give a characterization of all del Pezzo surfaces of degree 6 over an arbitrary field $F$. A surface is determined by a pair of separable algebras. These algebras are used to compute the Quillen $K$-theory of the surface. As a…

We classify indecomposable aCM bundles of rank $2$ on the del Pezzo threefold of degree $7$ and analyze the corresponding moduli spaces.

We classify the automorphism groups of del Pezzo surfaces of degrees one and two over an algebraically closed field of characteristic two. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.

This paper establishes the Manin conjecture for a certain non-split singular del Pezzo surface of degree four, via an analysis of the corresponding height zeta function.

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…

General hyperplane sections of a Fano threefold $Y$ of index 2 and Picard rank 1 are del Pezzo surfaces, and their Picard group is related to a root system. To the corresponding roots, we associate objects in the Kuznetsov component of $Y$…

The purpose of this note is to establish a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior. We identify unexpected plane curves of degree 4 as sections of a rational surface X of…