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.     

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.     

Hessian surfaces and local Lagrangian embeddings

In this paper we prove that any smooth surfaces can be locally isometrically embedded into ${\mathbb{C}}^2$ as Lagrangian surfaces. As a byproduct we obtain that any smooth surfaces are Hessian surfaces.     

Fermat-type configurations of lines in P3 and the containment problem

The purpose of this note is to show a new series of examples of homogeneous ideals I in $\mathbb{K}[x,y,z,w]$ for which the containment $I^{(3)}?I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb{P}}^3$, which resemble Fermat configurations of points in ${\mathbb{P}}^2$, see [14]. All examples exhibiting the failure of the containment $I^{(3)}?I^2$ constructed so far have been supported on points or cones over configurations of points. Apart from providing new counterexamples, these ideals seem quite interesting on their own.     

On the zeroth stable A1-homotopy group of a smooth curve

We provide a cohomological interpretation of the zeroth stable ${\mathbb{A}}^1$-homotopy group of a smooth curve over an infinite perfect field. We show that this group is isomorphic to the first Nisnevich (or Zariski) cohomology group of a certain sheaf closely related to the first Milnor–Witt K-theory sheaf. This cohomology group can be computed using an explicit Gersten-type complex. We show that if the base field is algebraically closed then the zeroth stable ${\mathbb{A}}^1$-homotopy group of a smooth curve coincides with the zeroth Suslin homology group that was identified by Suslin and Voevodsky with a relative Picard group. As a consequence we reobtain a version of Suslin's rigidity theorem.     

Global low-energy weak solution and large-time behavior for the compressible flow of liquid crystals

In this paper, we consider the weak solution of the simplified Ericksen–Leslie system modeling compressible nematic liquid crystal flows in ${\mathbb{R}}^3$. When the initial data are of small energy and initial density is positive and essentially bounded, we prove the existence of a global weak solution in ${\mathbb{R}}^3$. The large-time behavior of a global weak solution is also established.     

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.