Related papers: Degree of constantly curved holomorphic 2-spheres …

We complete Mori's program for Kontsevich's moduli space of degree 2 stable maps to Grassmannian of lines. We describe all birational models in terms of moduli spaces (of curves and sheaves), incidence varieties, and Kirwan's partial…

For a given manifold $M$ we consider the non-linear Grassmann manifold $Gr_n(M)$ of $n$-dimensional submanifolds in $M$. A closed $(n+2)$-form on $M$ gives rise to a closed 2-form on $Gr_n(M)$. If the original form was integral, the 2-form…

A submanifold of a Riemannian symmetric space is called parallel if its second fundamental form is a parallel section of the appropriate tensor bundle. We classify parallel submanifolds of the Grassmannian $\rmG^+_2(\R^{n+2})$ which…

For $n\in\{2^t-3,2^t-2,2^t-1\}$ ($t\ge3$) we study the cohomology algebra $H^*(\widetilde G_{n,3};\mathbb Z_2)$ of the Grassmann manifold $\widetilde G_{n,3}$ of oriented $3$-dimensional subspaces of $\mathbb R^n$. A complete description of…

We study the problem of determining the isomorphism classes of the virtually cyclic subgroups of the n-string braid groups B_n(S^2) of the 2-sphere S^2. If n is odd, or if n is even and sufficiently large, we obtain the complete…

The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\text{IG}(2, 2n)$. We show that these rings are regular. In particular, by "generic smoothness", we obtain a conceptual proof of generic…

The present paper is a natural continuation of a previous work where we studied the second syzygy scheme of canonical curves. We find sufficient conditions ensuring that the second syzygy scheme of a genus--$g$ curve of degree at least…

The bisymplectic Grassmannian I$_2$Gr$(k, V)$ parametrizes k-dimensional subspaces of a vector space V which are isotropic with respect to two general skew-symmetric forms; it is a Fano variety which admits an action of a torus with a…

In this note we consider homogeneous Willmore surfaces in $S^{n+2}$. The main result is that a homogeneous Willmore two-sphere is conformally equivalent to a homogeneous minimal two-sphere in $S^{n+2}$, i.e., either a round two-sphere or…

In this note we study morphisms from P2 to Gr(2,C4) from the point of view of the cohomology class they represent in the Grassmannian. This leads to some new result about projection of d-uple imbedding of P2 to P5.

In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both $\mathbb{C}P^n$ and the hyperquadric of $\mathbb{C}P^n$. The moduli space of all those noncongruent…

We study the integral Chow ring of the stack $\mathcal{H}_{g,n}$ parametrizing $n$-pointed smooth hyperelliptic curves of genus $g$. We compute the integral Chow ring of $\mathcal{H}_{g,n}$ for $n=1,2$ completely, while for $3\leq…

In projective space over fields of characteristic different from 2, the normal bundle of a general nondegenerate rational curve is balanced. The corresponding statement for rational curves in other Grassmannians can fail. Nevertheless, we…

We show that self-dual 2-forms in 2n dimensional spaces determine a $n^2-n+1$ dimensional manifold ${\cal S}_{2n}$ and the dimension of the maximal linear subspaces of ${\cal S}_{2n}$ is equal to the (Radon-Hurwitz) number of linearly…

We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.

We use recent developments by Gromov and Zhu to derive an upper bound for the 2-systole of the homology class of S 2 x { * } in a S 2 x S 2 with a positive scalar curvature metric such that the set of spheres homologous to S 2 x { * } is…

We give equations for 13 genus-2 curves over $\overline{\mathbb{Q}}$, with models over $\mathbb{Q}$, whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the…

For $r\geq 1$, the $r$-matching complex of a graph $G$, denoted $M_r(G)$, is a simplicial complex whose faces are the subsets $H \subseteq E(G)$ of the edge set of $G$ such that the degree of any vertex in the induced subgraph $G[H]$ is at…

We show that for every complete Riemannian surface $M$ diffeomorphic to a sphere with $k \geq 0$ holes there exists a Morse function $f:M \rightarrow \mathbb{R}$, which is constant on each connected component of the boundary of $M$ and has…

The present article studies holomorphic isometric embeddings of the complex two--plane Grassmannnian into quadrics. We discuss the moduli space of these embeddings up to gauge and image equivalence using a generalisation of do…