Classifies real rational knots and curves in a specific quadric space.
problem Classifying real rational knots and curves in a quadric space of signature (3,2). method Classification through a study of real rational curves of low degree in the quadric.
result Provides representatives of all real rational knots of degree ≤5 in the quadric. Three methods solve spatial rational curves with rational arc length.
problem Construct all spatial rational curves with rational arc length.
method Three different methods: PH curve adaptation, zero-residue conditions, and dual approach.
result Three methods share quaternion-based representation.
Survey on minimal rational curves and their geometric structures.
problem Germ-equivalence problem of minimal rational curves on uniruled projective manifolds.
method Analysis of isotrivial families of projective varieties and G-structures.
result Natural G-structure on Zariski-open subset of uniruled projective manifolds.
New findings on prime theta-curves with simple tangles.
problem Understanding prime theta-curves with specific unknotting numbers.
method Analyzing composite theta-curves and their components.
result Composite theta-curves with unknotting number one are prime.
Survey on rational curves on complex surfaces, highlighting different approaches.
problem Existence of rational curves on complex surfaces.
method Classification of complex surfaces and systematic study of rational curves in each class.
result Highlighting the different approaches to study rational curves on complex surfaces.
A conjecture about rational curves' formal principle and convergence proved for Goursat type families.
problem Formal principle and convergence for rational curves of Goursat type.
method Natural ODEs and Cartan connections constructed by Doubrov-Komrakov-Morimoto.
result The conjecture is proved for rational curves of Goursat type.
Study pseudo-hyperkähler geometry of curves in hyperkähler twistor spaces.
problem Understanding the geometry of rational curves in twistor spaces.
method Investigate pseudo-hyperkähler geometry of higher degree rational curves.
result Characterize the pseudo-hyperkähler structure of rational curves.
The study of symplectic fillings for rational cuspidal curves.
problem Understanding symplectic fillings of contact manifolds associated with rational cuspidal curves.
method Exploration through Stein handlebodies and rational blow-downs.
result Examples of contact manifolds that are links of normal surface singularities, and those that do not admit symplectic fillings.
Study of rational curves in complex manifolds with specific normal bundles.
problem Characterizing rational curves in complex manifolds with given normal bundles.
method Analyzing differential and projective geometric properties of rational curves and their tangents.
result Classification of rational curves into Goursat and Cartan types based on their geometric properties.
We construct normal rationally connected varieties (of arbitrarily large dimension) not containing any smooth rational curves.
New proof for curved 3-cohom manifold rational ellipticity.
problem Rational ellipticity of curved manifolds with specific cohomogeneity.
method Proved rationally elliptic for cohomogeneity-three manifolds with positive curvature and no boundary quotient.
result Closed, simply connected, positively curved, cohomogeneity-three manifolds without boundary are rationally elliptic.
Formula conjectured for rational cuspidal curves in projective plane.
problem Counting rational cuspidal curves in projective plane.
method Extending Kontsevich's recursion formula and using geometric input about tangency of curves at nodal points.
result Conjectural formula agrees with earlier computations and extends to rational quartics with E6 singularity.
Study delta invariant of curves on rational surfaces using topological methods.
problem Calculate delta invariant for curves embedded in rational singularities.
method Use topological techniques and Poincaré series.
result Develop formulae for delta invariant in terms of embedded data.
Classifies curves up to symplectic isotopy.
problem Classifying rational cuspidal curves up to symplectic isotopy.
method Topological tools, pseudoholomorphic techniques, and birational transformations.
result Classifies rational cuspidal curves of degrees 6 and 7 up to symplectic isotopy.
We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree.
Study minimal rational curves on complex manifolds with isotropic VMRT.
problem Understanding minimal rational curves tangent to distributions on complex manifolds.
method Partial equivariant compactification of metabelian groups.
result Any isotropic VMRT can be realized as VMRT of minimal rational curves tangent to a distribution.
We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form …
We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex proje…
Complex projective manifolds without rational curves are quotients of Abelian varieties.
problem Characterizing complex projective manifolds without rational curves.
method Using conjectures about rational and entire curves on Calabi-Yau varieties.
result Non-hyperbolic complex projective manifolds contain the image of an Abelian variety.
Formula counts rational curves with a specific singular point in projective space.
problem Counting rational degree d curves with an m-fold point in CP2. method Recursive formula derived from Kontsevich's recursion formula, considering a family version.
result Obtained a recursive formula for the number of curves.
By using analytic method, we prove that there exist rational curves on compact Hermitian manifolds with positive holomorphic bisectional curvature. It confirms a question of S.-T. Yau. It is well-known that Mori proved in \cite{Mori79} that every compact complex manifold N with c1(N)>0 contains at least one ration…
We explore a relationship between topological properties of orbits of 2-cycles in the symplectomorphism group Symp(M) and the existence of rational curves in M. Under the absence of rational curves hypothesis, we show that evaluation map vanishies on the second homotopy group and obtain a Gottlieb-type vanishing theore…
We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer theory and is a generalization of a result by Livingston and the first author. …
The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…
In this paper we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space o…
This paper proves an upper limit on rational points on curves.
problem Finding rational points on curves of genus at least two.
method Arithmetic and analytic estimates.
result Explicit upper bounds on rational points.
Study delta invariant of minimal generic curves on rational surfaces.
problem Recover delta invariant of curve germs from surface singularity topology.
method Explicit formulae for minimal generic curves on rational surfaces, proving delta invariant values for quotient singularities.
result Explicit formulae and values for delta invariant of minimal generic curves on rational surfaces.
Minimal rational curves on compactified symmetric spaces are orbit-closures of 1-parameter subgroups.
problem When are minimal rational curves on equivariant compactifications of symmetric spaces orbit-closures of 1-parameter subgroups?
method Combining algebraic geometry of minimal rational curves with differential geometry of symmetric spaces, showing Gauss-nondegeneracy of VMRT.
result The Gauss-nondegeneracy of VMRT implies that minimal rational curves on equivariant compactifications of symmetric spaces are orbit-closures of 1-parameter subgroups.
Solves open problems on curved projective varieties.
problem Structure theorems for curved projective varieties.
method Supplements and proposes open problems.
result Provides new insights into structure of curved projective varieties.
The study constructs balanced and rigid curves on specific types of hypersurfaces and complete intersections.
problem Constructing balanced and rigid curves on Calabi-Yau and general-type complete intersections.
method Balanced and rigid curves are constructed using specific hypersurfaces and complete intersections.
result Rigid curves of various genera and balanced rational curves of high degrees are constructed.
We find a general solution to the unique 7th order ODE admitting ten dimensional group of contact symmetries. The integral curves of this ODE are rational contact curves in $\PP^3$ which give rise to rational plane curves of degree six. The moduli space of these curves is a real form of the homogeneous space $Sp(4)/SL(…
We prove that any compact Kähler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact Kähler manifold.
In this paper we study non-negatively curved and rationally elliptic GKM4 manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifol…
We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows us to settle the existence and uniqueness problem. The machinery of dual polygons…
Classifies Fano varieties with large pseudoindex and non-free rational curves.
problem Classifying Fano varieties with specific properties.
method Extremal contractions and classification of varieties.
result Complete classification of Fano n-folds with pseudoindex at least n−2 and Picard number greater than one. The paper classifies sextic curves on a Fano 3-fold with rational Galois covers in 3D space.
problem Classifying sextic curves on a Fano 3-fold with rational Galois covers in 3D space.
method Explicit constructions and invariants of Galois groups.
result The moduli space of such sextic curves has complex dimension 2.
A method is proposed to construct spiral curves by inversion of a spiral arc of parabola. The resulting curve is rational of 4-th order. Proper selection of the parabolic arc and parameters of inversion allows to match a wide range of boundary conditions, namely, tangents and curvatures at the endpoints, including thos…
New method detects projective equivalences and symmetries in rational 3D curves.
problem Detecting projective equivalences and symmetries in rational 3D curves.
method Using differential invariants and Möbius transformations to avoid solving large polynomial systems.
result Efficient algorithm for detecting projective equivalences and symmetries without solving large polynomial systems.
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…
New proof shows rationality of scl for non-filling curves.
problem Understanding stable commutator length in non-filling curves.
method New proof using extremal surfaces for scl.
result Rationality of stable commutator length for non-filling curves.
The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn out to provide an infinite dimensional family of integrable systems.
Machine learning classifies complex geometric patterns with high accuracy.
problem Classifying extension degree of dessins d'enfants over the rationals.
method Deep feed-forward neural network trained on machine learning.
result 0.92 accuracy in classification with 0.03 standard error.
Rational configurations in K3 surfaces and simply-connected pg=1 surfaces for K2=1,2,3,4,5,6,7,8,9math.AG The study finds families of surfaces with specific properties and connects them to rational curve configurations in K3 surfaces.
problem Existence and properties of surfaces with specific canonical and geometric genus conditions.
method Study of rational curve configurations and use of Q-Gorenstein smoothings. result Existence of (20−2K2)-dimensional families of simply-connected surfaces with pg=1 and K2=1,2,3,4,5,6,7,8,9. New Stein fillings found for rational surface singularities.
problem Exploring Stein fillings of rational surface singularities.
method Using planar open books and Lefschetz fibrations, describe Stein fillings via symplectic disk arrangements.
result Many rational singularities admit Stein fillings not diffeomorphic to Milnor fibers.
Characterizes mappings preserving Pythagorean-hodograph curves.
problem Preserving Pythagorean-hodograph curves in various dimensions.
method Proves conformal functions with square rational dilation are PH-preserving.
result Conformal functions with square rational dilation are the only PH-preserving mappings.
Approximate 3D elastic curves with exact constraints
problem Designing and approximating 3D elastic curves
method Numerically stable method for recovering 11 parameters
result Fast and stable approximation of arbitrary curves
This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n-3. We define the Deligne-Mumford compactification of this space and tropical ψ-class divisors.
New construction shows VMRTs of unbendable curves can be Legendrian.
problem Characterize VMRTs of unbendable rational curves under contact structures.
method Used geometry of contact lines and symplectic geometry of distributions.
result VMRTs of Legendrian submanifolds can be realized.