Related papers: On free arrangements of three conics
The main purpose of this paper is to provide combinatorial constraints on the constructability of free and nearly free arrangements of smooth plane conics admitting certain ${\rm ADE}$ singularites.
We provide a classification result on nearly free arrangements of lines in the complex projective plane with nodes and triple points.
In the present paper, we study conic-line arrangements having nodes, tacnodes, and ordinary triple points as singularities. We provide combinatorial constraints on such arrangements and we give the complete classification of free…
In the present paper, we study arrangements of smooth plane conics having only nodes and tacnodes as the singularities. We provide an interesting estimation on the number of nodes and tacnodes that depends only on a linear function of the…
We study the geometry of $\mathcal{Q}$-conic arrangements in the complex projective plane. These are arrangements consisting of smooth conics and they admit certain quasi-homogeneous singularities. We show that such $\mathcal{Q}$-conic…
The main purpose of the present paper is to provide a partial classification, performed with respect the weak-combinatorics, of free arrangements consisting of lines and one smooth conic with quasi-homogeneous ordinary singularities.
In the present note we study combinatorial and algebraic properties of cubic-line arrangements in the complex projective plane admitting nodes, ordinary triple and $A_{5}$ singular points. We deliver a Hirzebruch-type inequality for such…
In this paper we study plus-one generated arrangements of conics and lines in the complex projective plane with simple singularities. We provide several degree-wise classification results that allow us to construct explicit examples of such…
We describe the behaviour of a free reduced plane projective curve with respect to the deletion, respectively addition, of a smooth conic. These results apply in particular to conic-line arrangements. We present some obstructions to the…
In the present note we provide a partial classification of nearly free conic-line arrangements in the complex plane having nodes, tacnodes, and ordinary triple points. In this setting, our theoretical bound tells us that the degree of such…
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 show that there are only finitely many combinatorial types of free real line arrangements with only double, triple and quadruple intersection points, and we enlist all admissible weak-combinatorics of them. Then we classify all real…
We construct new examples of free curve arrangements in the complex projective plane using point-line operators recently defined by the second author. In particular, we construct a new example of a conic-line arrangement with ordinary…
By the famous ADE classification rational double points are simple. Rational triple points are also simple. We conjecture that the simple normal surface singularities are exactly those rational singularities, whose resolution graph can be…
In this article, we study the effects of topological and smooth obstructions on the existence of rational homology complex projective planes that admit quotient singularities of small indices. In particular, we provide a classification of…
Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane…
We present a local classification of smooth projective surfaces in 3-space via projective transformations in accordance with singularity types of central projections up to codimension 4. We also discuss relations between our classification…
We prove that, on a large cone containing the constant multiplicities, the only free multiplicities on the braid arrangement are those identified in work of Abe, Nuida, and Numata (2009). We also give a conjecture on the structure of all…
The main goal of this note is to begin a systematic study on conic-line arrangements in the complex projective plane. We show a de Bruijn-Erd\H{o}s-type inequality and Hirzebruch-type inequality for a certain class of conic-line…
This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and…