Related papers: Quartic unexpected curves and surfaces
The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in Algebraic Geometry. In general, it is still an open problem to understand when the points fail…
In a recent paper arXiv:1602.02300v2, Cook II, Harbourne, Migliore and Nagel related the splitting type of a line arrangement in the projective plane to the number of conditions imposed by a general fat point of multiplicity $j$ to the…
Starting with the ground-breaking work of Cook II, Harbourne, Migliore and Nagel, there has been a lot of interest in unexpected hypersurfaces. In the last couple of months a considerable number of new examples and new phenomena has been…
Several papers have been written studying unexpected hypersurfaces. We say a finite set of points Z admits unexpected hypersurfaces if a general union of fat linear subspaces imposes less that the expected number of conditions on the ideal…
In the paper we present new examples of unexpected varieties. The research on unexpected varieties started with a paper of Cook II, Harbourne, Migliore and Nagel and was continued in the paper of Harbourne, Migliore, Nagel and Teitler. Here…
The purpose of this note is to establish a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior. We identify unexpected plane curves of degree 4 as sections of a rational surface X of…
The notion of an unexpected curve in the plane was introduced in 2018, and was quickly generalized in several directions in a flurry of mathematical activity by many authors. In this expository paper we first describe some of the main…
We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreduciblity of unexpected plane curves of a set of points $Z$ in $\mathbb{P}^2$, using the minimal degree of a Jacobian syzygy of the…
In a recent paper by Cook, et al., which introduced the concept of unexpected plane curves, the focus was on understanding the geometry of the curves themselves. Here we expand the definition to hypersurfaces of any dimension and, using…
We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an…
We introduce one of the most beautiful algebraic varieties known, a quintic hypersurface in projective five-space, which is invariant under the action of the Weyl group of $E_6$. This variety is intricately related with many other moduli…
Felix Klein in course of his study of the regular icosahedron and its symmetries encountered a highly symmetric configuration of $60$ points in ${\mathbb P}^3$. This configuration has appeared in various guises, perhaps post notably as the…
We study the congruence of bitangent lines of an irreducible surface in the 3-dimensional projective space in arbitrary characteristic, with special attention to quartic surfaces with rational double points and, in particular, Kummer…
We consider characterizations of projective varieties in terms of their tangents. S. Mori established the characterization of projective spaces in arbitrary characteristic by ampleness of tangent bundles. J. Wahl characterized projective…
We discuss eight new(?) configuration theorems of classical projective geometry in the spirit of the Pappus and Pascal theorems.
In this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties.…
We study the reciprocal position of nine points in the plane, according to their collinearities. In particular, we consider the case in which the nine points are contained in an irreducible cubic curve and we give their classification. If…
The aim of this note is to give a generalization of some results concerning unexpected hypersurfaces. Unexpected hypersurfaces occur when the actual dimension of the space of forms satisfying certain vanishing data is positive and the…
Unexpected hypersurfaces arise when vanishing in points of a set $Z$ and higher-order vanishing along a general linear subspace fails to impose the expected number of independent conditions on forms of a fixed degree. The phenomenon was…
In this paper, we categorize all isomorphism classes of quasi-elliptic surfaces over a field $k$ of characteristic 2 or 3. For every quasi-elliptic surface $X$, we classify all possible sequences of blow-downs from $X$ to the projective…