Fibres of embedding dimension 2 in generic projections

On the size and local equations of fibres of general projections

For a general birational projection of a smooth nondegenerate projective $n$-fold from $\mathbb P^{n+c}$ to $\mathbb P^m$, $n<m\leq(n+c)/2$, all fibres have total length asymptotically bounded by $2^{\sqrt{n}+1} $ and the fibres are locally defined by linear and quadratic equations.

Families of stable bundles on the fibres of the hyperkähler twistor projection

Given a holomorphic vector bundle $E$ on the twistor space $\mathrm{Tw}(M)$ of a simple hyperkähler manifold $M$, we view it as a family of bundles $\left\{E_I\right\}$ on the fibres $π^{-1}(I)$ of the twistor projection $π: \mathrm{Tw}(M) \to \mathbb{CP}^1$, and study the relationship between stability of $E$ and its fibrewise stability. We verify that the argument of Teleman establishing the Zariski openness of stability and semi-stability in families of bundles applies in the case of the family $\left\{E_I\right\}$. We prove a partial converse to a result of Kaledin and Verbitsky, showing t…

Unobstructedness of filling secants and the Gruson-Peskine general projection theorem

We prove an unobstructedness result for deformations of subvarieties constrained by intersections with another, fixed subvariety. We deduce smoothness and expected-dimension results for multiple-point loci of generic projections, mainly from a point or a line, or for fibres of embedding dimension 2 or less. The current version is a substantial enhancement of the previous one, including for the first time results on projection from a line.

cotangent bundle of complex projective space的等距嵌入与超kähler几何 via rank-1投影方案

Isometric Embeddings and Hyperkähler Geometry of the Cotangent Bundle of Complex Projective Space via the Scheme of Rank-1 Projections

本文通过rank-1投影方案研究cotangent bundle的超kähler几何，给出SU(n)等价的等距嵌入及超kähler结构在交换环上的推广。

Projections of Grassmannians of lines and characterization of Veronese varieties

We characterize the double Veronese embedding of P^n as the only variety that, under certain general conditions, can be isomorphically projected from the Grassmannian of lines in P^{2n+1} to the Grassmannian of lines in P^{n+1}.

Recovering an Algebraic Curve Using its Projections From Different Points. Applications to Static and Dynamic Computational Vision

We study how an irreducible closed algebraic curve X embedded in CP^3 can be recovered using its projections from points onto embedded projective planes. The different embeddings are unknown. The only input is the defining equation of each projected curve. We show how both the embeddings and the curve in CP^3 can be recovered modulo some actions of the group of projective transformations of CP^3. For two projections, we show how in a generic situation, a characteristic matrix of the two embeddings can be recovered. We also establish the minimal number of irreducible algebraic curves required t…