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T\^ete-\`a-t\^ete graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed t\^ete-\`a-t\^ete graphs provide a generalization which define mixed t\^ete-\`a-t\^ete twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed t\^ete-\`a-t\^ete twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed Dehn twists around disjoint simple closed curves, including all boundary components. It follows that the class of t\^ete-\`a-t\^ete twists coincides with that of monodromies associated with reduced function germs on isolated complex surface singularities., Comment: 21 pages, 14 figures. Minor corrections. Version as accepted in journal. arXiv admin note: text overlap with arXiv:1706.05580

We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom–Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of curves.

We show that a free surface in \(\mathbb {P}^3\) is characterized by the maximality of the degree of its singular subscheme, in the presence of an additional tameness condition. This is similar to the characterization of free plane curves by the maximality of their global Tjurina number given by A. A. du Plessis and C.T.C. Wall. Simple characterizations of the nearly free tame surfaces are also given.

We consider a class of Dupin hypersurface in $$R^5$$ parametrized by lines of curvature, with four distinct principal curvatures. We provide a characterization of such hypersurfaces in terms of the principal curvatures and three vector valued functions of one variable. We prove that these functions describe plane curves. The Lie curvature of these hypersurfaces is not constant but some Moebius curvatures are constant along certain lines of curvature. We give explicit examples of such Dupin hypersurfaces.

We study the caustic, evolute, Minkowski symmetry set and parallels of a smooth and regular curve in the Minkowski plane.

Enomoto, Weiner and the first author showed the rigidity of the Clifford torus amongst the class of embedded flat tori in S 3. In the proof of that result, an estimate of extrinsic diameter of flat tori plays a crucial role. It is reasonable to expect that the same rigidity holds in the class of immersed flat tori in S 3. In this paper, we give a new method for characterizing immersed flat tori in S 3 with extrinsic diameter π, which is a somewhat similar technique to the proof of the 6-vertex theorem for certain closed plane curves given by the second author. As an application, we show that the Clifford torus is rigid in the class of immersed flat tori whose mean curvature functions do not change sign. Recently, the global behaviour of flat surfaces in H 3 and R 3 regarded as wave fronts has been studied. We also give here a formulation of flat tori in S 3 as wave fronts. As an application, we shall exhibit a flat torus as a wave front whose extrinsic diameter is less than π.

The classical Whitney formula relates the number of times an oriented plane curve cuts itself to its rotation number and the index of a base point. In this paper we generalize Whitney's formula to curves on an oriented punctured surface. To define analogs of the rotation number and the index of a base point of a curve, we fix an arbitrary vector field on the surface. Similar formulas are obtained for non-based curves., Published version; 10 pages, 6 figures

In Finsler geometry, minimal surfaces with respect to the Busemann-Hausdorff measure and the Holmes-Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. In this paper, we give the explicit expressions of BH-minimal and HT-minimal rotational hypersurfaces generated by plane curves rotating around the axis in the direction of \({\tilde{\beta}^{\sharp}}\) in Minkowski (α, β)-space \({(\mathbb{V}^{n+1},\tilde{F_b})}\) , where \({\mathbb{V}^{n+1}}\) is an (n+1)-dimensional real vector space, \({\tilde{F_b}=\tilde{\alpha}\phi(\tilde{\beta}/\tilde{\alpha}), \tilde{\alpha}}\) is the Euclidean metric, \({\tilde{\beta}}\) is a one form of constant length \({b:=\|\tilde{\beta}\|_{\tilde{\alpha}}, \tilde{\beta}^{\sharp}}\) is the dual vector of \({\tilde{\beta}}\) with respect to \({\tilde{\alpha}}\) . As an application, we first give the explicit expressions of the forward complete BH-minimal rotational surfaces generated around the axis in the direction of \({\tilde{\beta}^{\sharp}}\) in Minkowski Randers 3-space \({(\mathbb{V}^{3},\tilde{\alpha}+\tilde{\beta})}\) .

We consider the following problem: For a smooth plane curve C of degree d ≥ 4 in characteristic p > 0, determine the number δ(C) of inner Galois points with respect to C. This problem seems to be open in the case where d ≡ 1 mod p and C is not a Fermat curve F(pe + 1) of degree pe + 1. When p ≠ 2, we completely determine δ(C). If p = 2 (and C is in the open case), then we prove that δ(C) = 0, 1 or d and δ(C) = d only if d−1 is a power of 2, and give an example with δ(C) = d when d = 5. As an application, we characterize a smooth plane curve having both inner and outer Galois points. On the other hand, for Klein quartic curve with suitable coordinates in characteristic two, we prove that the set of outer Galois points coincides with the one of \({\mathbb{F}_{2}}\) -rational points in \({\mathbb{P}^{2}}\).

Let C be a smooth irreducible projective curve of genus g and s(C, 2) (or simply s(2)) the minimal degree of plane models of C. We show the non-existence of curves with s(2) = g for g ≥ 10, g ≠ 11. Another main result is determining the value of s(2) for double coverings of hyperelliptic curves. We also give a criterion for a curve with big s(2) to be a double covering.

Using Dumnicki’s approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities in sufficiently general points on \({\mathbb{P}^2}\) we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on \({\mathbb{P}^2}\). With this method we prove the lower bound \({\frac{4}{13}}\) for 10 general points on \({\mathbb{P}^2}\).

For a smooth plane curve $$C \subset {\mathbf{P}}^2$$ , we call a point $$P \in {\mathbf{P}}^2$$ a Galois point if the point projection $$\pi_P:C \rightarrow {\mathbf{P}}^1$$ at P is a Galois covering. We study Galois points in positive characteristic. We give a complete classification of the Galois group given by a Galois point and estimate the number of Galois points for C in most cases.

In this paper, we study real hypersurfaces all of whose integral curves of characteristic vector fields are plane curves in a nonflat complex space form.

In 1974, Rokhlim introduced complex orientations for nonsingular real algebraic plane projective curves of type I. Here we give a definition of symmetric orientations and of "type" for T-curves which are PL-curves constructed using a combinatorial method called T-construction. An important aspect of T-construction is that, under particular conditions, the constructed T-curve has the isotopy type of a nonsingular real algebraic plane projective curve. T-construction is in fact a particular case of the method of construction of real algebraic projective varieties due to O. Ya. Viro. We prove that if an algebraic curve is associated to a T-curve by the Viro process, then the type of the T-curve coincides with the type of the algebraic curve and its symmetric orientations are complex orientations as defined by Rokhlin. The main result of this paper is the classification theorem for T-curves of type I.

If n is an odd prime less than 20, then the most symmetric nonsingular plane curves in P 2 of degree n are projectively equivalent to the Fermat curve x n +y n +z n .

First, a modern presentation of the theory of the Halphen transform is given. This method associates to a plane projective curve C, once a general conic has been chosen, another birationally equivalent plane curve, whose singularities are simpler than those of C. Repeating, a curve is obtained whose only singularities are nodes. Next, it is studied how to apply this process to a family of plane curves. With this technique it is possible to transform a given family (with irreducible general member) into one where, generically, the curves are nodal. Finally, it is studied a similar process, called the Halphen–Picard transformation, for surfaces in three-space. By suitably reiterating this procedure, a surface can be transformed into a birationally equivalent one (in the same projective space), such that the sections with planes in a general pencil are, generically, nodal curves.

After Gleason's result, in the late fifties the following conjecture appeared: if in a finite projective plane every quadrangle is contained in a unique Desarguesian proper subplane of order p, then the plane is Desarguesian (and its order is pd for some d). In this paper we prove the conjecture in the case when the plane is of order p2 and p is a prime.

Following a previous work of Tallini (J. Geom.\(29\) (1987), 191–199), we investigate the arithmetical properties of a blocking set in a finite projective plane which is met by any line in either 1 or k points, for a fixed number k. The results are then used to give some characterizations of blocking sets of this type which are preserved by a ‘large’ collineation group of the plane.

In this paper, we define the notion of the flex curve F(ℙ)(f; P) at a nonsingular point P of a plane curve Ca. We construct interesting plane curves using a cyclic covering transform, branched along F(ℙ)(f; P). As an application, we show the moduli space of projective curves of degree 12 with 27 cusps has at least three irreducible components. Simultaneously, we give an example of Alexander-equivalent Zariski pair of irreducible curves.

Our main result is that integrated geodesic curvature of a (nonsimple) closed curve on the unit two-sphere equals a half integer weighted sum of the areas of the connected components of the complement of the curve. These weights that gives a spherical analogy to the winding number of closed plane curves are found using Gauss–Bonnet’s theorem after cutting the curve into simple closed sub-curves. If the spherical curve is the tangent indicatrix of a space curve we obtain a new short proof of a formula for integrated torsion presented in an unpublished manuscript by C. Chicone and N. J. Kalton. Applying our result to the principal normal indicatrix we generalize a theorem by Jacobi stating that a simple closed principal normal indicatrix of a closed space curve with nonvanishing curvature bisects the unit two-sphere to nonsimple principal normal indicatrices. Some errors in the literature are corrected. We show that a factorization of a knot diagram into simple closed sub-curves defines an immersed disc with the knot as boundary and use this to give an upper bound on the unknotting number of knots.

It is shown that the group PSL2(H) cannot act effectively on any eight-dimensional stable plane. Together with previous results, this entails that every eight-dimensional stable plane admitting a nontrivial action of SL2(H) embeds into the projective plane over Hamilton's quaternions H.

The classification of cone-representations of projective planes of orderq3 of index 3 and rank 4 (and so in PG(6,q)) is completed. Any projective plane with a non-spread representation (being a cone-representation of the ‘second kind’) is a dual ‘generalised Desarguesian translation plane’, as found by Jha and Johnson, and conversely. Indeed, given any collineation of PG(2,q) with no fixed points, there exists such a projective plane of order q3 , where q is a prime power, that has the second kind of cone-representation of index 3 and rank 4 in PG(6,q). An associated semifield plane of order q3 is also constructed at most points of the plane. Although Jha and Johnson found this plane before, here we can show directly the geometrical connection between these two kinds of planes.

The total squared curvature of a closed plane curve is minimized by a covering of a circle, if the rotation number of the curve is not zero. We measure the closeness of a closed plane curve to a circle by the difference of its total squared curvature from the minimum. A similar result is obtained for a simple closed curve in the 2-sphere.

The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plucker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J + in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts.

A projective plane is called smooth if both the point space and the line space are smooth manifolds such that the geometric operations are smooth. We prove that every smooth projective translation plane is isomorphic to one of the classical planes over ℝ, ℂ, ℍ or\(\mathbb{O}\).

We are interested in a particular geometry of plane curves in characteristicp>0, which was inspired by Thas's article [13]. We will prove that any plane curve of degree > 2 whose tangent lines at collinear points are concurrent is either a strange curve or projectively equivalent to the Fermat curve of degreeq + 1, whereq is a power ofp.

If three Euclidean planes move relatively to each other, the three poles of rotation are either identical or pairwise distinct and collinear. In the second case the distances of the poles are in the ratio of the motions' angular velocities. These known facts of Euclidean kinematics can be generalized in a largely uniform way to plane Cayley/Klein motions with finite poles. For the angular velocities we give a representation which is valid for all considered Cayley/Klein motions. Application of the duality principle of the projective plane yields a proposition about concurrent fixed lines. We also generalize the generation of a pair of envelope curves of a Euclidean motion as paths of a point of a third moved plane.

Algorithms based on the theory of Delaney-Dress symbols are discussed that can be used to recursively produce all possible equivariant types of tile-k-transitive tilings of the Euclidean plane, the sphere and the hyperbolic plane, for any (reasonable)kel. A number of results can be obtained using computer implementations of the algorithms.

We investigate the problem of finding the maximum length of perimeters of plane sets with fixed diameter d, such that every point of the boundary of the set is a vertex of an open angle of opening α which does not intersect the set. First we consider plane curves which satisfy such angle property in a finite number of directions, and among them we find the one of maximum length. Then we prove that the perimeter of any plane set with the angle property is less than or equal to πd(sin α/2)-2; this is the best estimate when π/2≤α≤π.

A submanifold Mnr of Minkowski space \(\mathbb{E}_1^m \) is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of \(\mathbb{E}_1^m \) to the tangent space of Mnr at every point of Mnr. In this paper we completely classify hypersurfaces of restricted type in \(\mathbb{E}_1^{n + 1} \). More precisely, we prove that a hypersurface of \(\mathbb{E}_1^m \) is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: Sk × \(\mathbb{E}_1^{n - k} \), Sk1 × \(\mathbb{E}^{n - k} \), Hk × \(\mathbb{E}^{n - k} \), Sn1, Hn, with 1≤k≤n−1, or an open part of a cylinder on a plane curve of restricted type.

Here we prove that for every n≥33 and every t≤(n2+3n)/6, the normalization Y of a general plane curve C of degree n and with t nodes has no gb1 with b

This paper is devoted to the investigation of the geometry of left invariant metrics on the isometry group of the hyperbolic plane determined by the kinetic energy of a rigid particle system.

Spread sets of projective planes of order q3 are represented as sets of q3 points in A ≅ AG(3, q3). A line through the origin in A can be interpreted as a space A0 ≅ AG(3, q), and the spread set induces a cubic surface L in A0. If the projective plane is a semifield plane of dimension 3 over its kernel, then L has the property that it misses a plane of A0. Determining all such surfaces L leads to a complete classification of the semifield planes of order q3, whose spread sets are division algebras of dimension 3.

An affine or projective Hjelmslev plane (henceforth A.H. or P.H. plane) is a generalization of an ordinary affine or projective plane, where two lines (points) may intersect in (be joined by) more than one point (line). Two points are neighbours if they possess more than one joining line, and two lines are neighbours if each point of either line has a neighbour on the other. These neighbour relations define equivalence relations on the set of points and the set of lines respectively. Their corresponding quotient spaces generate an ordinary plane called the quotient plane of the Hjelmslev plane. A projective (affine) plane is a topological plane if the set of points and the set of lines are Hausdorff spaces, and the operations of join and intersection (and parallelism) are continuous. The purpose of this paper is to investigate analogous topological notions in Hjelmslev planes. Our ideas are naturally highly motivated by the work of Salzmann (cf. [11] and [12]). Every ordinary affine plane has a unique projective extension. This result has a topological analogue, provided the plane is Desarguesian or locally compact and connected (cf. [11], §14 and [12], 7.15). However, for H-planes, the author and N. D. Lane have constructed in [8] a Desarguesian A.H. plane with no Desarguesian projective extension, and in [3] Drake constructed an A.H. plane with no projective extension. Hence, we shall first study topological A.H. planes, in Sections 1 to 6, and then consider topological P.H. planes in Sections 7 and 8. Examples of such planes are constructed in Section 9. A P.H. (A.H.) plane is topological if its point and line sets are topological spaces, the neighbour relations are closed, and join and intersection (and parallelism) are continuous. Our main objectives are, to determine when the quotient plane, endowed with the usual quotient topologies, is a topological plane, and to construct examples of topological H-planes over certain topological rings. In Section 3 we determined relationships between point and line topologies and use these to prove join and intersection are open maps. Section 4 contains the result that the quotient plane is topological if and only if the quotient map is open on the points. If the plane is locally compact and connected or a translation plane, then the quotient plane is shown, in Section 5, to be * The author gratefully acknowledges the support of the National Research Council of Canada.

The closed plane curves of class C 2 which have curvature k(s) > 0 or k(s) ≥ 0 with a finite number of zeros are studied. The results concern the existence of normal lines which divide the perimeter into equal parts and the existence of some special kinds of pairs of points on these curves as orthodiameter pairs, antipodal pairs, etc. The paper also contains some generalizations of the theorems of Blaschke-Suss and Barbier.

A special Laguerre plane is a nondegenerate transversal 3-design such that the residue of each point is a dual affine plane. A special Laguerre plane is equivalent to an optimal code with three information digits and maximal length. An extended dual affine plane is an incidence structure (whose objects will be called points and blocks) such that the residue of each point is a dual affine plane, and each pair of points is in at least one block. Finite extended dual affine planes exist only of order 2, 4, and (dubiously) 10. We show that any finite incidence structure having the residue of each point a dual affine plane either is a transversal 3-design or has a block through each pair of points. Hence theorem: If a finite nondegenerate connected incidence structure θ has the residue of each point a dual affine plane, then θ is either an extended dual affine plane or a special Laguerre plane.

For ordered projective planes of Lenz Class at least II we give conditions for the existence of an order-preserving epimorphism onto an Archimedean projective plane. Every ordered plane of Lenz Class V allows such epimorphisms.

It is shown in this paper that a pair of points contained in a Fano configuration in a projective plane of odd order cannot induce a Minkowski plane. From this result we derive that no pair of points in the Hughes plane of order 9 can induce a Minkowski plane.

It is proved that a complete surface in E3 is a sphere or a plane if it contains at least four geodesics through each point which are plane curves.

The existence of Baer collineations in a projective plane is related to the existence of desargues-like configurations. The plane of order four is characterized as the only finite plane that possesses a Baer subplane partition into tangentially transitive Baer subplanes which is preserved by each of the tangentially transitive groups. It is shown that a finite projective plane has either no or one tangentially transitive Baer subplane or is partially transitive of Hughes type (4, m), (5, m) or (6, m) for some m. The Lenz-Barlotti classes which contain a finite plane which is not a translation plane nor its dual and which possesses a tangentially transitive Baer subplane are shown to be classes I.1 and II.1.