Related papers: On cubic surfaces with a rational line

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…

I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number…

We prove the sharp bound of at most 64 lines on complex projective quartic surfaces (resp. affine quartics) that are not ruled by lines. We study configurations of lines on certain non-K3 surfaces of degree four and give various examples of…

We study linear systems of surfaces in $\mathbb{P}^3$ singular along general lines. Our purpose is to identify and classify special systems of such surfaces, i.e., those nonempty systems where the conditions imposed by the multiple lines…

In this paper, we prove that a pair of the minimal resolution of a del Pezzo surface with rational double points whose general anti-canonical member is smooth and its exceptional divisor lifts to the Witt ring. We also classify a del Pezzo…

We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear $p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric…

We show that simultaneous log resolutions of simply elliptic singularities can be constructed inside suitable stacks of principal bundles over elliptic curves. In particular, we give a direct geometrical construction of del Pezzo surfaces…

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…

A parameterized surface can be represented as a projection from a certain toric surface. This generalizes the classical homogeneous and bihomogeneous parameterizations. We extend to the toric case two methods for computing the implicit…

We compute the constant of approximation for an arbitrary rational point on an arbitrary smooth cubic hypersurface $X$ over a number field $k$, provided that there is a $k$-rational line somewhere on $X$. In the process, we verify the Coba…

We study the density of everywhere locally soluble diagonal quadric surfaces, parameterised by rational points that lie on a split quadric surface.

We use a global version of Heath-Brown's $p-$adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most $B$ on non-singular cubic curves defined over $\mathbb{Q}$. The bounds are…

We construct a complex algebraic surface with geometric genus $p_g=3$, irregularity $q=0$, self-intersection of the canonical divisor $K^2=24$ and canonical map of degree $24$ onto $\mathbb P^2$.

We explicitly construct a component of the K-moduli space of K-polystable del Pezzo surfaces which is a smooth rational curve.

In this note, we establish an asymptotic formula for the number of rational points of bounded height on the singular cubic surface $$ x_0(x_1^2 + x_2^2)=x_3^3 $$ with a power-saving error term, which verifies the Manin-Peyre conjectures for…

Let S be a Dedekind scheme with fraction field K. We study the following problem: given a Del Pezzo surface X, defined over K, construct a distinguished integral model of X, defined over all of S. We provide a satisfactory answer if S is a…

We state conditions under which the set S(k) of k-rational points on a del Pezzo surface S of degree 1 over an infinite field k of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic…

We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for…

We construct orbits of the absolute Galois group, of explicit unbounded size, consisting of surfaces with mutually non-isomorphic fundamental groups. These are Beauville surfaces with Beauville group PGL_2(p).

In this paper, we analyze the Hessian locus associated to a general cubic hypersurface, by describing for every $n$ its singular locus and its desingularization. The strategy is based on strong connections between the Hessian and the…