Related papers: Counting lines on surfaces, especially quintics

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…

In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface $S$, and fix a number of points $F$ on its boundary. We ask: how many…

We study spaces of lines that meet a smooth hypersurface X in P^n to high order. As an application, we give a polynomial upper bound on the number of planes contained in a smooth degree d hypersurface in P^5 and provide a proof of a result…

We completely describe the structure of the connected components of transversals to a collection of n line segments in R^3. We show that n>2 arbitrary line segments in R^3 admit 0, 1, ..., n or infinitely many line transversals. In the…

Let $F(x_1,...,x_n)$ be a form of degree $d\geq 2$, which produces a geometrically irreducible hypersurface in $\mathbb{P}^{n-1}$. This paper is concerned with the number of rational points on F=0 which have height at most $B$. Whenever…

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…

We give a relatively short and elementary proof of Manin's conjecture for split smooth quintic del Pezzo surfaces over the rational numbers.

We study rational points on the elliptic surface given by the equation: $$y^2 = x^3 + AxQ(u,v)^2 + BQ(u,v)^3,$$ where $A,B\in \mathbb{Z}$ satisfy that $4A^3-27B^2\neq 0$ and $Q(u,v)$ is a positive-definite quadratic form. We prove…

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain $n \ge \b_2-2$ disjoint smooth rational curves with self-intersection -2, where $\b_2$ is the…

From a topological viewpoint, a rational curve in the real projective plane is generically a smoothly immersed circle and a finite collection of isolated points. We give an isotopy classification of generic rational quintics in…

A nonsingular surface of degree $d \geq 2$ in $\mathbb{P}^3$ over $\mathbb{F}_q$ has at most $((d-1)q+1)d$ $\mathbb{F}_q$-lines, and this bound is optimal for $d = 2, \sqrt{q}+1, q+1$.

In this paper we compute the number of rational curves with one node passing through a given number of points, lines and tangent to a given number of planes in $\mathbb{P}^3$.

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 consider a rational elliptic surface with a relatively minimal fibration. We compute the number of integral sections in the above rational elliptic surface. As an application, we obtain an estimate of polynomial solutions of some…

In this paper we provide, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e. Gauss curvature and mean curvature. In practice, the…

In this, the last of three papers about $C_2$-equivariant complex quadrics, we complete the calculation of the equivariant ordinary cohomology of smooth symmetric quadrics in the cases where the fixed sets have more than two components.…

We study qualitative aspects of the Welschinger-like $\mathbb Z$-valued count of real rational curves on primitively polarized real $K3$ surfaces. In particular, we prove that with respect to the degree of the polarization, at logarithmic…

We advance our understanding of the configurations of low degree smooth rational curves on (quasi-)polarized complex K3-surfaces. We apply our efficient approach to classify the configurations of at least 36 lines on K3-sextics with at…

We prove the following results: (1) For every generic closed smooth curve in $\mathbb{R}^3$ there is a point with at least $6$ emanating normals to the curve. (2) For every generic closed piecewise linear curve in $\mathbb{R}^3$ there is a…

We show that any rational cubic hypersurface of dimension at least 33 defined over a number field $K$ vanishes on a $K$-rational projective line, reducing the previous lower bound of Wooley by two. For $K=\mathbb Q$ we can reduce the bound…