Related papers: On trisecant lines to White surfaces

We prove that if a finite point set in real space does not have too many points on a plane, then it spans a quadratic number of ordinary lines. This answers the real case of a question of Basit, Dvir, Saraf, and Wolf. It shows that there is…

We study integral points on affine surfaces by means of a new method, relying on the Subspace Theorem. Under suitable assumptions on the divisor at infinity, we prove that the integral points are contained in a curve. As a corollary, we…

The nonsingular Hermitian surface of degree $\sqrt{q} +1$ is characterized by its number of $\Bbb{F}_q$-points among the irreducible surfaces over $\Bbb{F}_q$ of degree $\sqrt{q} +1$ in the projective 3-space.

Let ${\mathcal G}_k(V)$ be the $k$-Grassmannian of a vector space $V$ with $\dim V=n$. Given a hyperplane $H$ of ${\mathcal G}_k(V)$, we define in [I. Cardinali, L. Giuzzi, A. Pasini, A geometric approach to alternating $k$-linear forms, J.…

We prove that the half-integer valued local index of an isolated umbilic point on a $C^{3+\alpha}$-smooth convex surface in Euclidean 3-space is less than two. The approach is to study the co-kernel of an associated Riemann-Hilbert boundary…

We study the set of $D$ such that a given irreducible hypersurface $C$ of degree $d$ has infinitely many points of degree $D$ over $\mathbb{Q}$. We give a new explicit proof that this set contains all (positive) multiples of the index of…

In [BN] the authors construct a special complex of degree 20 over M, which for an open three dimensional set parametrizes smooth complex surfaces of degree four invariant which are Heisenberg invariant and each member of the family contains…

It is shown that the edge ring of a finite connected simple graph with a $3$-linear resolution is a hypersurface.

We study ramified covers of the projective plane. Given a smooth projective surface S and a generic enough projection of S to the projective plane, we get a cover of the plane ramified over a plane curve. The branch curve is usually…

A point P on a smooth hypersurface X of degree d in an N-dimensional projective space is called a star point if and only if the intersection of X with the embedded tangent space T_P(X) is a cone with vertex P. This notion is a…

We show that infinitely many Gorenstein weakly-exceptional quotient singularities exist in all dimensions, we prove a weak-exceptionality criterion for five-dimensional quotient singularities, and we find a sufficient condition for being…

We will use toric degenerations of the projective plane ${{\mathbb{P}}^ 2}$ to give a new proof of the triple points interpolation problems in the projective plane. We also give a complete list of toric surfaces that are useful as…

In the present paper, we discuss the singular minimal surfaces in a Euclidean 3-space R^{3} which are minimal. In fact, such a surface is nothing but a plane, a trivial outcome. However, a non-trivial outcome is obtained when we modify the…

Let $R$ be a polynomial ring over a field and $I$ an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension $\mathrm{pd}(R/I)$ of $R/I$ is at most 36, although the…

We consider the ring of invariants of n points on the projective line. The space (P^1)^n // PGL_2 is perhaps the first nontrivial example of a Geometry Invariant Theory quotient. The construction depends on the weighting of the n points.…

Suppose that a Hilbert scheme of points on a K3 surface S of Picard rank one admits a rational Lagrangian fibration. We show that if the degree of the surface is sufficiently large compared to the number of points, then the Hilbert scheme…

Given a compact Riemann surface $C$, the line in $H^0(J_C,\, 2\Theta)$ orthogonal to the sections vanishing at $0$ produces a natural projective structure on $C$. We investigate the properties of this projective structure.

We discuss the work of Corrado Segre on nodal cubic hypersurfaces with emphasis on the cases of 6-nodal and 10-nodal cubics. In particular, we discuss the Fano surface of lines and conic bundle structures on such threefolds. We review some…

Over the complex numbers, there are 92 plane conics meeting 8 general lines in projective 3-space. Using the Euler class and local degree from motivic homotopy theory, we give an enriched version of this result over any perfect field. This…

We prove that, for $q$ odd, a set of $q+2$ points in the projective plane over the field with $q$ elements has at least $2q-c$ odd secants, where $c$ is a constant and an odd secant is a line incident with an odd number of points of the…