Related papers: Rational surfaces in P^4 containing a plane curve

We define and study a family of cubic surfaces in the projectivized tangent bundle over a four dimensional projective space associated to the root system D_5. The 27 lines are rational over the base and we determine the classifying map to…

We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such…

We construct a surface of general type with canonical map of degree 12 which factors as a triple cover and a bidouble cover of $\mathbb P^2$. We also show the existence of a smooth surface with $q=0,$ $\chi=13$ and $K^2=9\chi$ such that its…

We discover a simple construction of a four-dimensional family of smooth surfaces of general type with $p_g(S)=q(S)=0$, $K^2_S=3$ with cyclic fundamental group $C_{14}$. We use a degeneration of the surfaces in this family to find…

This is a preliminary note on a family of minimal surfaces in the 3-sphere defined by a compatible fourth order equation. The minimal surfaces are geometrically characterized either by having a surface of revolution like induced metric, or…

We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler…

This is an addendum to the paper of Braun and Fl{\o}ystad ([BF]) on the bound for the degree of a smooth surface in $\pfour$ not of general type. Using their construction and the regularity of curves in $\pthree$, one may lower the bound a…

In this paper we study sets of points in the plane with rational distances from r prescribed points P_1, ...,P_r. A crucial case arises for r = 3, where we provide simple necessary and sufficient conditions for the density of this set in…

Let $X$ be a del Pezzo surface of degree $5$ defined over a field $F$. A theorem of Yu. I. Manin and P. Swinnerton-Dyer asserts that every Del Pezzo surface of degree $5$ is rational. In this paper we generalize this result as follows.…

We show that Brauer classes of a locally solvable degree 4 del Pezzo surface X are vertical for some projection away from a plane f: X ---> P^1, i.e., that every Brauer class is obtained by pullback from an element of Br k(P^1). As a…

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…

We prove that any smooth rational projective surface over the field of complex numbers has an open covering consisting of 3 subsets isomorphic to affine planes.

A rational map $\phi: \mathbb{P}^1 \to \mathbb{P}^1$ along with an ordered list of fixed and critical points is called a totally marked rational map. The space of totally marked degree two rational maps, $Rat^{tm}_2$ can be parametrized by…

For any given natural $d\ge 1$ we provide examples of rational self-maps of complex projective plane $\pp^2$ of degree $d$ without (holomorphic) fixed points. This makes a contrast with the situation in one dimension. We also prove that the…

In this note we show the existence of a family of elliptic conic bundles in P^4 of degree 8. This family has been overlooked and in fact falsely ruled out in a series of classification papers. Our surfaces provide a counterexample to a…

Given a morphism between smooth projective varieties $f: W \to X$, we study whether $f$-relatively free rational curves imply the existence of $f$-relatively very free rational curves. The answer is shown to be positive when the fibers of…

Let F be a finite field of characteristic p. We consider smooth surfaces over F(t) defined by an equation f+tg=0, where f and g are forms of degree d in 4 variables with coefficients in F, with d prime to p. We prove : For such surfaces…

The global geometry of the moduli spaces of higher spin curves and their birational classification is largely unknown for g >= 2 and r > 2. Using quite related geometric constructions, we almost complete the picture of the known results in…

Let $X$ be an algebraic variety, defined over the rationals. This paper gives upper bounds for the number of rational points on $X$, with height at most $B$, for the case in which $X$ is a curve or a surface. In the latter case one excludes…

In this article we construct a new family of simply connected symplectic 4-manifolds with $b_2^+ =1$ and $c_1^2 =2$ which are not diffeomorphic to rational surfaces by using rational blow-down technique. As a corollary, we conclude that a…