Related papers: Tropicalization of Del Pezzo Surfaces

In this article, we give the classification of normal del Pezzo surfaces of rank one with at most log canonical singularities containing the affine plane defined over an algebraically non-closed field of characteristic zero. As an…

We provide explicit graded constructions of orbifold del Pezzo surfaces with rigid orbifold points of type $\left\{k_i\times\frac{1}{r_i}(1,a_i): 3\le r_i \le 10,k_i \in \ZZ_{\ge 0}\right\}$; as well-formed and quasismooth varieties…

We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these…

We give constructions of completions of the affine $3$-space into total spaces of del Pezzo fibrations of every degree other than $7$ over the projective line. We show in particular that every del Pezzo surface other than $\mathbb{P}^{2}$…

Real, complex, and tropical algebraic geometry join forces in a new branch of mathematical physics called positive geometry. We develop the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties.…

We give a new geometric proof of the classification of $T$-polygons, a theorem originally due to Kasprzyk, Nill and Prince, using ideas from mirror symmetry. In particular, this gives a completely geometric proof that any two toric…

We revisit a correspondence between toroidal compactifications of M-theory and del Pezzo surfaces, in which rational curves on the del Pezzo are related to ${1\over 2}$-BPS branes of the corresponding compactification. We argue that curves…

We compute the coregularity of del Pezzo surfaces with du Val singularities. To this aim, we study the relation between del Pezzo surfaces of degree $1$ and elliptic fibrations. It turns out that del Pezzo surfaces with positive…

We give a complete classification of del Pezzo surfaces with quotient singularities and Picard rank 1 which admit a Q-Gorenstein smoothing. There are 14 infinite families of toric examples. The surfaces in each family correspond to…

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.

Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground fields to arbitrary non-archimedean valued fields. To achieve this, we develop…

We classify all the del Pezzo surfaces with $\frac{1}{3}(1,1)$- and $\frac{1}{4}(1,1)$-singularities having no floating $(-1)$-curves into 39 types.

Patchworking theorems serve as a basic element of the correspondence between tropical and algebraic curves, which is a core of the tropical enumerative geometry. We present a new version of a patchworking theorem which relates plane…

We look at how one can construct from the data of a dimer model a Lagrangian submanifold in $(\mathbb{C}^*)^n$ whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with…

We tropicalize the rational map that takes triples of points in the projective plane to the plane of quadrics passing through these points. The image of its tropicalization is contained in the tropicalization of its image. We identify these…

In this paper, we classify del Pezzo foliations on projective manifolds of rank at least 3 and with log canonical singularities in the sense of McQuillan.

Toric log Del Pezzo surfaces with Picard number 1 have been completely classified whenever their index is $\le 2$: In this paper we extend the classification for those having index 3: We prove that, up to isomorphism, there are exactly 18…

Let $f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$ and let us consider a del Pezzo surface of degree one given by the equation $\cal{E}_{f}: x^2-y^3-f(z)=0$. In this note we prove that if the set of rational points on the curve $E_{a,…

The blow-up of the anticanonical base point on a del Pezzo surface $S$ of degree 1 gives rise to a rational elliptic surface $\mathscr{E}$ with only irreducible fibers. The sections of minimal height of $\mathscr{E}$ are in correspondence…

We classify smooth weak del Pezzo surfaces with global vector fields over an arbitrary algebraically closed field $k$ of arbitrary characteristic $p \geq 0$. We give a complete description of the configuration of $(-1)$- and $(-2)$-curves…