Related papers: On surfaces in P^6 with no trisecant lines

Very few examples of obstructed equsingular families of curves on surfaces other than the projective plane are known. Combining results from Westenberger and Hirano with an idea from math.AG/9802009 we give in the present paper series of…

Motivated by our study (elsewhere) of linear syzygies of homogeneous ideals generated by quadrics and their restrictions to subvarieties of the ambient projective space, we investigate in this note possible zero-dimensional intersections of…

In this paper we compute upper bounds for the number of ordinary triple points on a hypersurface in $P^3$ and give a complete classification for degree six (degree four or less is trivial, and five is elementary). But the real purpose is to…

We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length…

Let $S$ be a smooth minimal surface of general type with a (rational) pencil of hyperelliptic curves of minimal genus $g$. We prove that if $K_S^2<4\chi(\mathcal O_S)-6,$ then $g$ is bounded. The surface $S$ is determined by the branch…

We obtain the sharp $l^p$ decoupling for three-dimensional nondegenerate surfaces in $\mathbb{R}^6$. This can be thought of as a generalization of Bourgain and Demeter's result, which is the sharp $l^p$ decoupling for two-dimensional…

We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.

This note (which makes no claim to novelty) presents a proof of the separable rational connectedness of smooth cubic hypersurfaces, in any characteristic, by showing how to explicitly construct very free curves (of degree 3) on them. -----…

We study projective surfaces in $\mathbb{P}^3$ which can be written as Hadamard product of two curves. We show that quadratic surfaces which are Hadamard product of two lines are smooth and tangent to all coordinate planes, and such…

If $S$ is a quintic surface in $\mathbb P^3$ with singular set $15$ $3$-divisible ordinary cusps, then there is a Galois triple cover $\phi:X\to S$ branched only at the cusps such that $p_g(X)=4,$ $q(X)=0,$ $K_X^2=15$ and $\phi$ is the…

In this article we classify all the smooth threefolds of P^5 with an apparent quadruple point provided that the family of its 4-secant lines is an irreducible (first order) congruence. This is sufficient to conclude the classification of…

We give a simple proof of T. Stehling's result, that in any normal tiling of the plane with convex polygons with number of sides not less than six, all tiles except the finite number are hexagons.

We consider the action of the group $\mathrm{PGL}_4(K)$ on the smooth cubic surfaces of $\mathbb{P}^3_K$ ($K$ an algebraically closed field of characteristic zero). We classify, in an explicit way, all the smooth cubic surfaces with non…

Let $S$ be a parametric surface in $\proj{3}$ given as the image of $\phi: \proj{1} \times \proj{1} \to \proj{3}$. This paper will show that the use of syzygies in the form of a combination of moving planes and moving quadrics provides a…

We prove that any triangulation of a surface different from the sphere and the projective plane admits an orientation without sinks such that every vertex has outdegree divisible by three. This confirms a conjecture of Bar\'at and Thomassen…

We prove that for any convex polygon $S$ with at least four sides, or a concave one with no parallel sides, and any $m>0$, there is an $m$-fold covering of the plane with homothetic copies of $S$ that cannot be decomposed into two…

We present a complete classification of complex projective surfaces $X$ with nontrivial self-maps (i.e. surjective morphisms $f:X\rightarrow X$ which are not isomorphisms) of any given degree. The starting point of our classification are…

We give quantitative and qualitative results on the family of surfaces in $\mathbb{CP}^3$ containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines $E$. We prove that its general element…

We examine the local geometry of affine surfaces which are locally symmetric. There are 6 non-isomorphic local geometries. We realize these examples as Type A, Type B, and Type C geometries using a result of Opozda and classify the relevant…

We prove some lower bounds on certain nonegative twists of the canonical bundle of a subvariety of a generic hypersurface in projective space. In particular we prove that the generic sextic threefold contains no rational or elliptic curves…