Related papers: Weak Approximation for Low Degree del Pezzo Surfac…

Over the function field of a complex algebraic curve, strong approximation off a non-empty finite set of places holds for the complement of a codimension $2$ closed subset in a homogeneous space under a semisimple algebraic group, and for…

We study surjective (not necessarily regular) rational endomorphisms $f$ of smooth del Pezzo surfaces $X$. We prove that under certain natural non\,-\,degeneracy condition $f$ can have degree bigger than $1$ only when $(-K_X^2) > 5$. Some…

We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be…

In this paper we obtain necessary and sufficient condition for existence of del Pezzo surfaces of degree $5$ and $6$ over a field $K$ with a prescribed action of absolute Galois group $\text{Gal} ( K^{\text{sep}}/K)$ on the graph of…

Let $P$ be a finite simplicial comple with underlying space (union of simplices in $P$) $|P|$. Let $Q$ be a subcomplex of $P$. Let $a \geq 0$. Then there exists $K < \infty$, \emph{depending only on $a$ and $Q$,} with the following…

Previous work of the authors showed that every quartic del Pezzo surface over a number field has index dividing $2$ (i.e., has a closed point of degree $2$ modulo $4$),, and asked whether such surfaces always have a closed point of degree…

In this paper, we study the family ${\mathcal C}_{H}^0$ of sense-preserving complex-valued harmonic functions $f$ that are normalized close-to-convex functions on the open unit disk $\mathbb{D}$ with $f_{\bar{z}}(0)=0$. We derive a…

Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we…

For a regular del Pezzo surface $X$, we prove that $|-12K_X|$ is very ample. Furthermore, we also give an explicit upper bound for the volume $K_X^2$ which depends only on $[k : k^p]$ for the base field $k$. As a consequence, we obtain the…

We complete the classification of local stability thresholds for smooth del Pezzo surfaces of degree~2. In particular, we show that this number is irrational if and only if a unique (-1)-curve passes through the point where we are computing…

Let S be a complete surface of constant curvature K = + 1 or -1, i.e. the sphere S^2 or the Lobachevskij plane L^2, and D a bounded convex subset of S. If S = S^2, assume also diameter (D) < pi/2. It is proved that the length of any…

Let $R$ be a semilocal principal ideal domain. Two algebraic objects over $R$ in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all…

We study points of moderately low degree on a curve $C$ over a number field, which is embedded on a nice toric surface $S$. Recently, Smith and Vogt related the linear equivalence classes of such points to intersections of $C$ with curves…

A projective threefold transition $Y \xrightarrow{\phi} \bar{Y} \rightsquigarrow X$ is del Pezzo if $\phi$ contracts a smooth del Pezzo surface to a point. We show that the GW/PT correspondence holds on $Y$ implies that it holds on $X$. In…

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…

For any fixed $1 \leq \ell \leq 9$, we characterize all Wahl singularities that appear in degenerations of del Pezzo surfaces of degree $\ell$. This extends the work of Manetti and Hacking-Prokhorov in degree $9$, where Wahl singularities…

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…

For Ch\^atelet surfaces defined over number fields, we study two arithmetic properties, the Hasse principle and weak approximation, when passing to an extension of the base field. Generalizing a construction of Y. Liang, we show that for an…

For a Del Pezzo surface of degree 8 given over the rationals we decide whether there is a rational parametrization of the surface and construct one in the affirmative case. We define and use the Lie algebra of the surface to reach the aim.…

We classify the number of $k$-rational lines and conic fibrations on del Pezzo surfaces over a field $k$ in terms of relatively minimal surfaces and establish rational curve analogues of the inverse Galois problem for del Pezzo surfaces. We…