Plane Conics Research Articles

Bridgeland stability conditions and skew lines on ℙ3

Inspired by Schmidt’s work on twisted cubics, we study wall crossings in Bridgeland stability, starting with the Hilbert scheme Hilb 2 m + 2 ( P 3 ) parametrizing pairs of skew lines and plane conics union a point. We find two walls. Each wall crossing corresponds to a contraction of a divisor in the moduli space and the contracted space remains smooth. Building on work by Chen–Coskun–Nollet, we moreover prove that the contractions are K-negative extremal in the sense of Mori theory and so the moduli spaces are projective.

On free and nearly free arrangements of conics admitting certain ADE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{ADE}}$$\\end{document} singularities

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The elliptic sieve and Brauer groups

AbstractA theorem of Serre states that almost all plane conics over have no rational point. We prove an analogue of this for families of conics parametrised by elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve for elliptic curves. We also give more general results for specialisations of Brauer groups, which yields applications to norm form equations.

Conics meeting eight lines over perfect fields

Over the complex numbers, there are 92 plane conics meeting 8 general lines in projective 3-space. Using the Euler number and local degree from motivic homotopy theory, we give an enriched version of this result over any perfect field. This provides a weighted count of the number of plane conics meeting 8 general lines, where the weight of each conic is determined the geometry of its intersections with the 8 given lines. As a corollary, real conics meeting 8 general lines come in two families of equal size.

Differential geometry of complex projective plane conics

Differential geometry of complex projective plane conics

On plane conic arrangements with nodes and tacnodes

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 number of conics. Based on that result, we obtain a new upper bound on the number of tacnodes which turns out to be better than Miyaoka's bound for a large enough number of conics. We also study the freeness and nearly freeness of such arrangements providing a detailed description.

A geometric invariant of 6-dimensional subspaces of 4×4 matrices

Let R R denote a 6-dimensional subspace of the ring M 4 ( k ) M_4(\Bbbk ) of 4 × 4 4 \times 4 matrices over an algebraically closed field k \Bbbk . Fix a vector space isomorphism M 4 ( k ) ≅ k 4 ⊗ k 4 M_4(\Bbbk ) \cong \Bbbk ^4 \otimes \Bbbk ^4 . We associate to R R a closed subscheme X R {\mathbf X}_R of the Grassmannian of 2-dimensional subspaces of k 4 \Bbbk ^4 , where the reduced subscheme of X R {\mathbf X}_R is the set of 2-dimensional subspaces Q ⊆ k 4 Q \subseteq \Bbbk ^4 such that ( Q ⊗ k 4 ) ∩ R ≠ { 0 } (Q \otimes \Bbbk ^4) \cap R \ne \{ 0\} . Our main result is that if X R {\mathbf X}_R has minimal dimension (namely, one), then its degree is 20 when it is viewed as a subscheme of P 5 \mathbb {P}^5 via the Plücker embedding. We present several examples of X R \mathbf X_R that illustrate the wide range of possibilities for it; there are reduced and non-reduced examples. Two examples involve elliptic curves: in one case, X R {\mathbf X}_R is a P 1 \mathbb {P}^1 -bundle over an elliptic curve the second symmetric power of the curve; in the other, it is a curve having seven irreducible components, three of which are quartic elliptic space curves, and four of which are smooth plane conics. These two examples arise naturally from a problem having its roots in quantum statistical mechanics. The scheme X R \mathbf X_R appears in non-commutative algebraic geometry: under appropriate hypotheses, it is isomorphic to the line scheme L \mathcal {L} of a certain graded algebra determined by R R . In that context, it has been an open question for several years to describe such L \mathcal {L} of minimal dimension, i.e., those L \mathcal {L} of dimension one. Our main result implies that if dim ⁡ ( L ) = 1 \dim (\mathcal {L}) = 1 , then, as a subscheme of P 5 \mathbb {P}^5 under the Plücker embedding, deg ⁡ ( L ) = 20 \deg (\mathcal {L}) = 20 .

The decomposition groups of plane conics and plane rational cubics

The decomposition group of an irreducible plane curve $X\subset\mathbb P^2$ is the subgroup $\mathrm{Dec}(X)\subset\mathrm{Bir}(\mathbb P^2)$ of birational maps which restrict to a birational map of $X$. We show that $\mathrm{Dec}(X)$ is generated by its elements of degree $\leq2$ when $X$ is either a conic or rational cubic curve.

Minimum weight codewords in dual algebraic-geometric codes from the Giulietti-Korchmáros curve

In this paper we investigate the number of minimum weight codewords of some dual algebraic-geometric codes associated with the Giulietti–Korchmaros maximal curve, by computing the maximal number of intersections between the Giulietti–Korchmaros curve and lines, plane conics, and plane cubics.

Exotic elliptic algebras of dimension 4

Let E be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3. Let τ be a translation automorphism of E that is not of order 2. In a previous paper we studied an algebra A=A(E,τ) that depends on this data: A(E,τ)=(S(E,τ)⊗M2(k))Γ where S(E,τ) is the 4-dimensional Sklyanin algebra associated to (E,τ), M2(k) is the ring of 2×2 matrices over k, and Γ is (Z/2)×(Z/2) acting in a particular way as automorphisms of S and M2(k). The action of Γ on S is compatible with the translation action of the 2-torsion subgroup E[2] on E. Following the ideas and results in papers of Artin–Tate–Van den Bergh, Smith–Stafford, and Levasseur–Smith, this paper examines the line modules, point modules, and fat point modules, over A, and their incidence relations. The right context for the results is non-commutative algebraic geometry: we view A as a homogeneous coordinate ring of a non-commutative analogue of P3 that we denote by Projnc(A). Point modules and fat point modules determine “points” in Projnc(A). Line modules determine “lines” in Projnc(A). Line modules for A are in bijection with certain lines in P(A1⁎)≅P3 and therefore correspond to the closed points of a certain subscheme L of the Grassmannian G(1,3). Shelton–Vancliff call L the line scheme for A. We show that L is the union of 7 reduced and irreducible components, 3 quartic elliptic space curves and 4 plane conics in the ambient Plücker P5, and that deg⁡(L)=20. The union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in P(A1⁎). Thus, the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for A.

Equivariant and invariant theory of nets of conics with an application to Thom polynomials

Two parameter families of plane conics are called nets of conics. There is a natural group action on the vector space of nets of conics, namely the product of the group reparametrizing the underlying plane, and the group reparametrizing the parameter space of the family. We calculate equivariant fundamental classes of orbit closures. Based on this calculation we develop the invariant theory of nets of conics. As an application we determine Thom polynomials of contact singularities of type (3,3). We also show how enumerative problems in particular the intersection multiplicities of the determinant map from nets of conics to plane cubics can be solved studying equivariant classes of orbit closures.

A new construction of anti-self-dual four-manifolds

We describe a new construction of anti-self-dual metrics on four-manifolds. These metrics are characterized by the property that their twistor spaces project as affine line bundles over surfaces. To any affine bundle with the appropriate sheaf of local translations, we associate a solution of a second-order partial differential equations system D2V = 0 on a five-dimensional manifold \({\mathbf{Y}}\). The solution V and its differential completely determine an anti-self-dual conformal structure on an open set in {V = 0}. We show how our construction applies in the specific case of conformal structures for which the twistor space \({\mathcal{Z}}\) has \({ \dim\left|-\frac{1}{2}K_\mathcal{Z}\right|\geq 2}\), projecting thus over \({\mathbb C\mathbb P_2}\) with twistor lines mapping onto plane conics.

Towards relative invariants of real symplectic four-manifolds

Let (X, ω, c X ) be a real symplectic four-manifold with real part \(\mathbb{R}X\). Let \(L \subset \mathbb{R}X\) be a smooth curve such that \([L] = 0 \in H_1 (\mathbb{R}X;\mathbb{Z}/2\mathbb{Z}).\) We construct invariants under deformation of the quadruple (X, ω, c X , L) by counting the number of real rational J-holomorphic curves which realize a given homology class d, pass through an appropriate number of points and are tangent to L. As an application, we prove a relation between the count of real rational J-holomorphic curves done in [W2] and the count of reducible real rational curves done in [W3]. Finally, we show how these techniques also allow us to extract an integer valued invariant from a classical problem of real enumerative geometry, namely about counting the number of real plane conics tangent to five given generic real conics.

Simultaneous and Sequential Reconstruction of Visual Primitives with Bundle Adjustment

In this paper, we compare the various methods for the simultaneous and sequential reconstruction of points, lines, planes, quadrics, plane conics and degenerate quadrics using Bundle Adjustment, both in projective and metric space. In contrast, most existing work on projective reconstruction focuses mainly on one type of primitive. We also compare the simultaneous refinement of all primitives through Bundle Adjustment with various sequential methods were only certain primitives are refined together. We found that even though the sequential methods may seem somewhat arbitrary on the choice of which primitives are refined together, a higher precision and speed is achieved in most cases.

The projective reconstruction of points, lines, quadrics, plane conics and degenerate quadrics using uncalibrated cameras

In this paper we present an algorithm for the simultaneous projective reconstruction of points, lines, quadrics, plane conics and degenerate quadrics using Bundle Adjustment. In contrast, most existing work on projective reconstruction focuses mainly on one type of primitive. Furthermore, for the reconstruction of quadrics (both full-rank and degenerate) and plane conics, a novel algorithm for the rectification of the outlines projected in n views is presented. Finally, we found out that due to the noise in the data, the global minimum of the classic cost function used in Bundle Adjustment may unfortunately break the topology of the quadrics. Hence, during the iterations of this method, we need to add the constraints to keep the quadrics within acceptable limits to preserve their topology.

The number of plane conics that are five-fold tangent to a given curve

The number of plane conics that are five-fold tangent to a given curve

Common Transversals and Tangents to Two Lines and Two Quadrics in P

We solve the following geometric problem, which arises in several {\mbox three-dimensional} applications in computational geometry: For which arrangements of two lines and two spheres in ${\Bbb R}^3$ are there infinitely many lines simultaneously transversal to the two lines and tangent to the two spheres? We also treat a generalization of this problem to projective quadrics. Replacing the spheres in ${\bf R}^3$ by quadrics in projective space ${\Bbb P}^3$, and fixing the lines and one general quadric, we give the following complete geometric description of the set of (second) quadrics for which the two lines and two quadrics have infinitely many transversals and tangents: in the nine-dimensional projective space ${\Bbb P}^9$ of quadrics, this is a curve of degree 24 consisting of 12 plane conics, a remarkably reducible variety.

Real rational curves in Grassmannians

Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some given general figures [5]. For the problem of plane conics tangent to five general (real) conics, the surprising answer is that all 3264 may be real [13]. Similarly, given any problem of enumerating p-planes incident on some given general subspaces, there are real subspaces such that each of the (finitely many) incident p-planes is real [17]. We show that the problem of enumerating parameterized rational curves in a Grassmannian satisfying simple (codimension 1) conditions may have all of its solutions real. This problem of enumerating rational curves in a Grassmannian arose in at least two distinct areas of mathematics. The number of such curves was predicted by the formula of Vafa and Intriligator [20, 8] from mathematical physics. It is also the number of complex dynamic compensators which stabilize a given linear system, and the enumeration was solved in this context [12, 11]. The question of real solutions also arose in systems theory [3]. This application will be discussed in Section 4.