This chapter reviews the recent debate about the role of social capital in economics. We argue that all the difficulties this concept has encountered in economics are due to a vague and excessively broad definition. For this reason, we restrict social capital to the set of values and beliefs that help cooperation - which for clarity we label civic capital. We argue that this definition differentiates social capital from human capital and satisfies the properties of the standard notion of capital. We then argue that civic capital can explain why differences in economic performance persist over centuries and discuss how the effect of civic capital can be distinguished empirically from other variables that affect economic performance and its persistence, including institutions and geography.

Civic Capital as the Missing Link

The European Proceedings of Social & Behavioural Sciences

We generalize the theory of Horrocks monads to ACM varieties, and use the generalization to establish a cohomological characterization of linear and Steiner bundles on projective space and on quadric hypersurfaces. We also characterize Steiner bundles on the Grassmannian G(1, 4) of lines in ℙ4. Finally, we study linear resolutions of bundles on ACM varieties, and characterize linear homological dimension on quadric hypersurfaces.

Linear and Steiner Bundles on Projective Varieties

We use a generalization of Horrocks monads for arithmetic Cohen-Macaulay (ACM) varieties to establish a cohomological characterization of linear and Steiner bundles over projective spaces and quadric hypersurfaces. We also study resolutions of bundles on ACM varieties by line bundles, and characterize linear homological dimension in the case of quadric hypersurfaces.

Linear and Steiner bundles on projective varieties

Virtual classes and virtual motives of Quot schemes on threefolds

Abstract A vector bundle E on a smooth irreducible algebraic variety X is called a Steiner bundle of type (F 0, F 1) if it is defined by an exact sequence of the form where s, t ≥ 1 and (F 0, F 1) is a strongly exceptional pair of vector bundles on X such that is generated by global sections. Let X be a smooth irreducible projective variety of dimension n with an n-block collection , of locally free sheaves on X which generate Db (𝒪 X –mod). We give a cohomological characterisation of Steiner bundles of type on X, where 0 ≤ a < b ≤ n and 1 ≤ i 0 ≤ αa , 1 ≤ j 0 ≤ αb .

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Cohomological characterisation of Steiner bundles

We study Steiner bundles on the smooth hyperquadric Qn ⊂ ℙn+1, for n ≥ 3. These are vector bundles E defined by an exact sequence of the form We characterize generic exceptional and generic simple Steiner bundles and we prove that generic exceptional Steiner bundles are stable.

Steiner Bundles on the Hyperquadric Q n ⊂ ℙ n+1, n ≥ 3

A very long-standing problem in Algebraic Geometry is to determine the stability of exceptional vector bundles on smooth projective varieties. In this paper we address this problem and we prove that any exceptional vector bundle on a smooth complete intersection 3-fold Y C P" of type (d 1 ;.., d n-3 ) with α i +... + d n-3 < n and n ≥ 4 is stable.

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The stability of exceptional bundles on complete intersection 3-folds

We generalise Floystad's theorem on the existence of monads on the projective space to a larger set of projective varieties. We consider a variety $X$, a line bundle $L$ on $X$, and a base-point-free linear system of sections of $L$ giving a morphism to the projective space whose image is either arithmetically Cohen-Macaulay (ACM), or linearly normal and not contained in a quadric. We give necessary and sufficient conditions on integers $a$, $b$, and $c$ for a monad of type \[ 0\to(L^\vee)^a\to\mathcal{O}_{X}^{\,b}\to L^c\to0 \] to exist. We show that under certain conditions there exists a monad whose cohomology sheaf is simple. We furthermore characterise low-rank vector bundles that are the cohomology sheaf of some monad as above. 
Finally, we obtain an irreducible family of monads over the projective space and make a description on how the same method could be used on an ACM smooth projective variety $X$. We establish the existence of a coarse moduli space of low-rank vector bundles over an odd-dimensional $X$ and show that in one case this moduli space is irreducible.

/pdf/monads-on-projective-varieties-y8ijy6ytff.pdf

Monads on projective varieties

Let X be an n-dimensional smooth projective variety with an n-block collection , with , of coherent sheaves on X that generate the bounded derived category . We give a cohomological characterisation of torsion-free sheaves on X that are the cohomology of monads of the form 






where . We apply the result to get a cohomological characterisation when X is the projective space, the smooth hyperquadric or the Fano threefold V5. We construct a family of monads on a Segre variety and apply our main result to this family.

Helena Soares

Papers

Cohomological characterisation of Steiner bundles

Steiner Bundles on the Hyperquadric Q n ⊂ ℙ n+1, n ≥ 3

The stability of exceptional bundles on complete intersection 3-folds

Monads on projective varieties

Cohomological characterisation of monads