Nonlinear estimates for hypersurfaces in terms of their fundamental forms

A sufficiently regular hypersurface immersed in the $(n+1)$-dimensional Euclidean space is determined up to a proper isometry of ${\mathbb{R}}^{n+1}$ by its first and second fundamental forms. As a consequence, a sufficiently regular hypersurface with boundary, whose position and positively-oriented unit normal vectors are given on a non-empty portion of its boundary, is uniquely determined by its first and second fundamental forms. We establish here stronger versions of these uniqueness results by means of inequalities showing that an appropriate distance between two immersions from a domain ω of ${\mathbb{R}}^n$ into ${\mathbb{R}}^{n+1}$ is bounded by the $L^p$-norm of the difference between matrix fields defined in terms of the first and second fundamental forms associated with each immersion.     

Rank two aCM bundles on the del Pezzo fourfold of degree 6 and its general hyperplane section

In the present paper we completely classify locally free sheaves of rank 2 with vanishing intermediate cohomology modules on the image of the Segre embedding ${\mathbb{P}}^2\times {\mathbb{P}}^2?{\mathbb{P}}^8$ and its general hyperplane sections.Such a classification extends similar already known results regarding del Pezzo varieties with Picard numbers 1 and 3 and dimension at least 3.     

A1-homotopy groups,excision, and solvable quotients

We study some properties of ${\mathbb{A}}^1$-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of ${\mathbb{A}}^1$-homotopy theory. These concepts and results are well suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of solvable group quotients, we investigate ${\mathbb{A}}^1$-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree ${\mathbb{A}}^1$-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the “next” non-vanishing ${\mathbb{A}}^1$-homotopy group (beyond ${\pi }_1^{{\mathbb{A}}^1}$) of a smooth toric variety. From this point of view, ${\mathbb{A}}^1$-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost “as tractable” (in low degrees) as ordinary homotopy for large classes of interesting varieties.     

On two upper bounds for hypersurfaces involving a Thas’ invariant

Let $X^n$ be a hypersurface in ${\mathbb{P}}^{n+1}$ with $n\ge 1$ defined over a finite field ${\mathbb{F}}_q$ of $q$ elements. In this note, we classify, up to projective equivalence, hypersurfaces $X^n$ as above which reach two elementary upper bounds for the number of ${\mathbb{F}}_q$-points on $X^n$ which involve a Thas’ invariant.     

The minimal measurement number for low-rank matrix recovery

The paper presents several results that address a fundamental question in low-rank matrix recovery: how many measurements are needed to recover low-rank matrices? We begin by investigating the complex matrices case and show that $4nr?4r^2$ generic measurements are both necessary and sufficient for the recovery of rank-r matrices in ${\mathbb{C}}^{n\times n}$. Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound $4nr?4r^2$ is tight provided $n=2^k+r,k\in {\mathbb{Z}}_{+}$. Motivated by Vinzant's work [19], we construct 11 matrices in ${\mathbb{R}}^{4\times 4}$ by computer random search and prove they define injective measurements on rank-1 matrices in ${\mathbb{R}}^{4\times 4}$. This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than $2n?1$ orthogonal projections are possible for the recovery of $x\in {\mathbb{R}}^n$ from the norm of them, which gives a negative answer for a question raised in [1].     

Stable Ulrich bundles on Fano threefolds with Picard number 2

In this paper, we consider the existence problem of rank one and two stable Ulrich bundles on imprimitive Fano 3-folds obtained by blowing-up one of ${\mathbb{P}}^3$, Q (smooth quadric in ${\mathbb{P}}^4$), $V_3$ (smooth cubic in ${\mathbb{P}}^4$) or $V_4$ (complete intersection of two quadrics in ${\mathbb{P}}^5$) along a smooth irreducible curve. We prove that the only class which admits Ulrich line bundles is the one obtained by blowing up a genus 3, degree 6 curve in ${\mathbb{P}}^3$. Also, we prove that there exist stable rank two Ulrich bundles with $c_1=3H$ on a generic member of this deformation class.