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. Let q∈C, let \[a=\begin{pmatrix} 1&0\\1&1\end{pmatrix},\quad b_q=\begin{pmatrix} 1&q\\0&1\end{pmatrix},\] and let Gq<SL2(C) be the group generated by a and bq. In this paper, we study the problem of determining when the group Gq is not free for ∣q∣<4 rational. We give a robu…
The study shows that surface groups are the only non-free infinite index subgroups of certain hyperbolic groups.
problem Identifying the only non-free infinite index subgroups of specific hyperbolic and one-relator groups.
method Careful analysis of free and cyclic splittings of cubulated groups.
result Proves that surface groups are the only non-free infinite index subgroups of certain hyperbolic and one-relator groups.
Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence, Kollár formulated the algebraic version of the Montgomery-Yang problem: every projective surface S with the second Betti number $b_2…
Wextend the results obtained recently by G. D'Ambra and A. Loi towards the proof of a conjecture of M.Gromov on isometric immersions via non-free maps.
Article explores non-freeness of groups generated by two specific matrices, providing counterexamples and sequences.
problem Tackles the non-freeness of groups generated by two parabolic matrices with rational parameters.
method Uses the orbit test and modulo homomorphisms to provide sufficient conditions and counterexamples.
result Constructs explicit counterexamples and sequences converging to 3, demonstrating non-freeness.
The study classifies manifolds realized as orbit spaces of non-free Z2^k actions.
problem Classifying manifolds realized as orbit spaces of non-free Z2^k actions.
method Examining actions of subgroups H on real moment-angle manifolds and analyzing orbit spaces.
result Constructs series of manifolds homeomorphic to S^n and manifolds admitting hyperelliptic involutions.
Agol's announcement proved a full classification of certain Kleinian groups.
problem Classifying non-free Kleinian groups generated by two parabolic transformations.
method Full proof of Agol's announcement.
result Classification of non-free Kleinian groups generated by two parabolic transformations.
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.
Suppose G is a non-free finitely generated Kleinian group without parabolics which is not a lattice and let C(G) denote the commensurator in PSL(2,C). We prove that if the limit set of G is not a round circle, then C(G) is discrete. Furthermore, G has finite index in C(G) unless G is a fiber group in which case C(G) is…
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. …
Study bounds and asymptotic behavior of largest purely non-free actions on compact Riemann surfaces.
problem Understanding the largest purely non-free actions on compact Riemann surfaces.
method Analyzing continuous actions of finite groups on closed orientable surfaces, proving bounds and asymptotic behavior.
result The largest order of a purely non-free action on a surface of even genus is bounded below by 8g and sharp for infinitely many even g.
We prove a conjecture of Gromov about non-free isometric immersions.
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 on cr-invariant variational problem for Legendrian curves in 3-sphere.
problem Lower-order cr-invariant variational problem for Legendrian curves in 3-sphere.
method Deduced Euler-Lagrange equations, investigated closed critical curves, characterized non-constant cr-curvature curves, proved cr-equivalence classes correspondence to rational points.
result Closed critical curves with non-constant cr-curvature are characterized and their cr-equivalence classes are in one-to-one correspondence with rational points of a connected planar domain.
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 delta invariant of curves on rational surfaces is calculated using embedded topological and analytic methods.
problem Calculating the delta invariant of curves on rational surfaces.
method Embedded topological and analytic approaches.
result The delta invariant can be recovered with a concrete expression associated with the embedded topological type of the pair (X,C).
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(…
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 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.
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…
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.