Complex Projective Plane Research Articles

Projective plane curves whose automorphism groups are simple and primitive

We study complex projective plane curves with a given group of automorphisms. Let $G$ be a simple primitive subgroup of $\mathrm{PGL}(3, \mathbf{C})$, which is isomorphic to $\mathfrak{A}_{6}$, $\mathfrak{A}_{5}$ or $\mathrm{PSL}(2, \mathbf{F}_{7})$. We obtain a necessary and sufficient condition on $d$ for the existence of a nonsingular projective plane curve of degree $d$ invariant under $G$. We also study an analogous problem on integral curves.

On the Circumcenters of Triangular Orbits in Elliptic Billiard

On an elliptic billiard, we study the set of the circumcenters of all triangular orbits and we show that this is an ellipse. This article follows Romaskevich (L’Enseig Math 60:247–255, 2014), which proves the same result with the incenters, and Glutsyuk (Moscow Math J 14:239–289, 2014), which among others, introduces the theory of complex reflection in the complex projective plane. The result we present was found at the same time in Garcia (Amer Math Monthly 126:491–504, 2019). His proof uses completely different methods of real differential calculus.

On 3-syzygy and unexpected plane curves

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. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

Hirzebruch-Type Inequalities Viewed as Tools in Combinatorics

The main purpose of this survey is to provide an introduction, algebro-topological in nature, to Hirzebuch-type inequalities for plane curve arrangements in the complex projective plane. These inequalities gain more and more interest due to their utility in many combinatorial problems related to point or line arrangements in the plane. We would like to present a summary of the technicalities and also some recent applications, for instance in the context of the Weak Dirac Conjecture. We also advertise some open problems and questions.

Subgroups of elliptic elements of the Cremona group

Abstract The Cremona group is the group of birational transformations of the complex projective plane. In this paper we classify its subgroups that consist only of elliptic elements using elementary model theory. This yields in particular a description of the structure of torsion subgroups. As an application, we prove the Tits alternative for arbitrary subgroups of the Cremona group, generalizing a result of Cantat. We also describe solvable subgroups of the Cremona group and their derived length, refining results from Déserti.

Minimal Lagrangian surfaces in [formula omitted] via the loop group method Part I: The contractible case

In this paper, we employ the loop group method to study the construction of minimal Lagrangian surfaces in the complex projective plane for which the domain of the surface is contractible. We present several new classes of minimal Lagrangian surfaces in ℂP2.

Genera of knots in the complex projective plane

Our goal is to systematically compute the [Formula: see text]-genus of as many prime knots up to 8-crossings as possible. We obtain upper bounds on the [Formula: see text]-genus via coherent band surgery. We obtain lower bounds by obstructing homological degrees of potential slice discs. The obstructions are pulled from a variety of sources in low-dimensional topology and adapted to [Formula: see text]. There are 27 prime knots and distinct mirrors up to 7-crossings. We now know the [Formula: see text]-genus of all of these knots. There are 64 prime knots and distinct mirrors up to 8-crossings. We now know the [Formula: see text]-genus of all but 6 of these knots, where the [Formula: see text]-genus was not determined explicitly, it was narrowed down to 2 possibilities. As a consequence of this work, we show an infinite family of knots such that the [Formula: see text]-genus of each knot differs from that of its mirror.

An inequality for length and volume in the complex projective plane

We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer--Mrowka proof of the Thom conjecture.

SMOOTH, NONSYMPLECTIC EMBEDDINGS OF RATIONAL BALLS IN THE COMPLEX PROJECTIVE PLANE

Abstract We exhibit an infinite family of rational homology balls, which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson’s diagonalization theorem and use this to show that no two of our examples may be embedded disjointly.

On the Hilbert vector of the Jacobian module of a plane curve

We identify several classes of complex projective plane curves C:f=0 , for which the Hilbert vector of the Jacobian module N(f) can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible components. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on \mathbb P^2 , is used to get a sharp lower bound for the initial degree of the Jacobian module N(f) , under a semistability condition.

Singularities and genus of the k-ellipse

A k-ellipse is a plane curve consisting of all points whose distances from k fixed foci sum to a constant. We determine the singularities and genus of its Zariski closure in the complex projective plane. The paper resolves an open problem stated by Nie, Parrilo and Sturmfels in 2008.

FOUR-MANIFOLDS WITH POSITIVE CURVATURE

AbstractIn this note, we prove that a four-dimensional compact oriented half-conformally flat Riemannian manifold M4 is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2}$ , provided that the sectional curvatures all lie in the interval $\left[ {{{3\sqrt {3 - 5} } \over 4}, 1} \right]$ In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the 4-sphere.

Newton–Okounkov bodies of exceptional curve valuations

We prove that the Newton–Okounkov body associated to the flag E_{\bullet}:= \{ X=X_r \supset E_r \supset \{q\} \} , defined by the surface X and the exceptional divisor E_r given by any divisorial valuation of the complex projective plane \mathbb{P}^2 , with respect to the pull-back of the line-bundle \mathcal{O}_{\mathbb{P}^2} (1) is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton–Okounkov bodies which turn out to be triangular.

Topology and homotopy of lattice isomorphic arrangements

We prove the existence of lattice isomorphic line arrangements having π 1 \pi _1 -equivalent or homotopy-equivalent complements and non-homeomorphic embeddings in the complex projective plane. We also provide two explicit examples: one is formed by real-complexified arrangements, while the second is not.

On Seshadri constants and point-curve configurations

<p style='text-indent:20px;'>In the note we study the multipoint Seshadri constants of <inline-formula><tex-math id="M1">$ \mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1) $</tex-math></inline-formula> centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of Seshadri constants can be approximated with use of a combinatorial invariant which we call the configurational Seshadri constant. We study specific examples of point-curve configurations for which we provide actual values of the associated Seshadri constants. In particular, we provide an example based on Hesse point-conic configuration for which the associated Seshadri constant is computed by a line. This shows that multipoint Seshadri constants are not purely combinatorial.</p>

On the Multipliers at Fixed Points of Quadratic Self-Maps of the Projective Plane with an Invariant Line

This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely determined, up to linear equivalence, by the collection of these eigenvalues. Under the natural correspondence between quadratic rational maps of $\mathbb{P}^2$ and quadratic homogeneous vector fields on $\mathbb{C}^3$, the algebraic relations among multipliers translate to algebraic relations among the Kowalevski exponents of a vector field. As an application of our results, we describe the sets of integers that appear as the Kowalevski exponents of a class of quadratic homogeneous vector fields on $\mathbb{C}^3$ having exclusively single-valued solutions.

Plane curves with three syzygies, minimal Tjurina curves, and nearly cuspidal curves

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.

Real hypersurfaces with constant ϕ-sectional curvature in complex projective space

We will give a geometric description of real hypersurfaces with constant ϕ-sectional curvature in complex projective space. Besides geodesic hypersurfaces, such real hypersurfaces are obtained as the image of either a curve or a surface in complex projective space under the polar map. As a consequence, we obtain a classification of real hypersurface M3 in complex projective plane such that the structure vector ξ is an eigenvector of the Ricci tensor.

On the minimal value of global Tjurina numbers for line arrangements

We show that a general lower bound for the global Tjurina number of a reduced complex projective plane curve, given by Andrew A. du Plessis and Charles T. C. Wall, can be improved when the curve is a line arrangement. This fact is in sharp contrast to a conjecture saying that the general upper bound for the global Tjurina number of a reduced complex projective plane curve, also given by du Plessis and Wall, is realized by line arrangements in practically all cases.

Convex foliations of degree 4 on the complex projective plane

We show that up to automorphism of $${\mathbb {P}}^{2}_{{\mathbb {C}}}$$ there are 5 homogeneous convex foliations of degree four on $${\mathbb {P}}^{2}_{{\mathbb {C}}}.$$ Using this result, we give a partial answer to a question posed in 2013 by Marin and Pereira about the classification of reduced convex foliations on $${\mathbb {P}}^{2}_{{\mathbb {C}}}.$$