Irreducibility of Lagrangian Quot schemes over an algebraic curve

Let C be a complex projective smooth curve and W a symplectic vector bundle of rank 2n over C. The Lagrangian Quot scheme $$LQ_{-e}(W)$$ L Q - e ( W ) parameterizes subsheaves of rank n and degree $$-e$$ - e which are isotropic with respect to the symplectic form. We prove that $$LQ_{-e}(W)$$ L Q - e ( W ) is irreducible and generically smooth of the expected dimension for all large e, and that a generic element is saturated and stable.

Baltiskie vietvārdu elementi

After the differentiation of the Indo-European languages, Baltic languages continued to exist side by side and develop their common geographical lexis. In the Baltic common lexis, mainly hyponyms or specific object names have formed, mostly words of the Eastern Baltic (i. e. Latvian and Lithuanian) origin. The paper deals with the most widespread generic terms of Baltic – mostly Curonian or Lithuanian – origin in Latvian toponymy: kalva ‘hillock’, danga ‘uneven place’, banda ‘field given by an owner to a servant’, lanka ‘wet meadow’, dzira ‘forest’, cērps ‘mound’, krants ‘shore, bank’, lincis ‘bay’, puosums ‘clearance’, sāts ‘meadow; populated place’, viņģis ‘bay’, vanga ‘flood-land’. In most cases, they have very branched polysemy, for example, danga 1) ‘beaten track’, 2) ‘uneven place’, 3) ‘piece of land which is surrounded by swamp or water from three sides’, 4) ‘entry of a building or forest’, 5) ‘corner’, 6) ‘bank’, 7) ‘pot on the road’. It seems that the generic terms of Curonian origin have broadened their meaning in Latvian much more than Lithuanianisms. The analysis of the most widespread generic elements of Baltic origin shows that specific toponymic system has been developed in the Western part of Latvia. Mostly borrowings from Curonian (lanka, cērps, krants, lincis, puosums, sāts) and Lithuanian (kalva, danga, viņģis) languages make these differences more distinct. The generic element vanga that until now has been considered a Curonianism, probably is an appellative of Finno-Ugric origin.

On the arithmetic of a family of degree - two K3 surfaces

Let ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x, y, z and w; let $\mathcal{X}$ be the generic element of the family of surfaces in ℙ given by \begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface $\mathcal{X}$ is a K3 surface over the function field ℚ(t). In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$, together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the family X.

Geometric Characterizations of Hilbert Spaces

We study some geometric properties related to the setobtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also compute the distance of a generic element (h, k) ∊ for H a Hilbert space.

Sand Piles Models of Signed Partitions with Piles

Let be nonnegative integers. In this paper we study the basic properties of a discrete dynamical model of signed integer partitions that we denote by . A generic element of this model is a signed integer partition with exactly all distinct nonzero parts, whose maximum positive summand is not exceeding and whose minimum negative summand is not less than . In particular, we determine the covering relations, the rank function, and the parallel convergence time from the bottom to the top of by using an abstract Sand Piles Model with three evolution rules. The lattice was introduced by the first two authors in order to study some combinatorial extremal sum problems.

Simple wild -packets

In a recent paper, Gross and Reeder study the arithmetic properties of discrete Langlands parameters for semi-simple$p$-adic groups, and they conjecture that a special class of these – the simple wild parameters – should correspond to$L$-packets consisting of simple supercuspidal representations. We provide a construction of this correspondence, and show that the simple wild$L$-packets satisfy many expected properties. In particular, they admit a description in terms of the Langlands dual group, and contain a unique generic element for a fixed Whittaker datum. Moreover, we prove their stability on an open subset of the regular semi-simple elements, and show that they satisfy a natural compatibility with respect to unramified base-change.

NON-EMPTINESS OF MODULI SPACES OF COHERENT SYSTEMS

Let X be a general smooth projective algebraic curve of genus g ≥ 2 over ℂ. We prove that the moduli space G(α:n,d,k) of α-stable coherent systems of type (n,d,k) over X is empty if k > n and the Brill–Noether number β := β(n,d,n + 1) = β(1,d,n + 1) = g - (n + 1)(n - d + g) < 0. Moreover, if 0 ≤ β < g or β = g, n ∤g and for some α > 0, G(α : n,d,k) ≠ ∅ then G(α : n,d,k) ≠ ∅ for all α > 0 and G(α : n,d,k) = G(α′ : n,d,k) for all α,α′ > 0 and the generic element is generated. In particular, G(α : n,d,n + 1) ≠ ∅ if 0 ≤ β ≤ g and α > 0. Moreover, if β > 0 G(α : n,d,n + 1) is smooth and irreducible of dimension β(1,d,n + 1). We define a dual span of a generically generated coherent system. We assume d < g + n1≤ g + n2and prove that for all α > 0, G(α : n1,d, n1+ n2) ≠ ∅ if and only if G(α : n2,d, n1+ n2) ≠ ∅. For g = 2, we describe G(α : 2,d,k) for k > n.