Chern Classes of Tautological Sheaves on Hilbert Schemes

We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on surfaces within the framework of Nakajima's oscillator algebra. This leads to an identification of the cohomology ring of Hilbert schemes of the affine plane with a ring of differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilb^n up to n=7, and give a conjecture for the generating series.Comment: 45 pages, LaTe

Vulnerability analysis of three remote voting methods

This article analyses three methods of remote voting in an uncontrolled environment: postal voting, internet voting and hybrid voting. It breaks down the voting process into different stages and compares their vulnerabilities considering criteria that must be respected in any democratic vote: confidentiality, anonymity, transparency, vote unicity and authenticity. Whether for safety or reliability, each vulnerability is quantified by three parameters: size, visibility and difficulty to achieve. The study concludes that the automatisation of treatments combined with the dematerialisation of the objects used during an election tends to substitute visible vulnerabilities of a lesser magnitude by invisible and widespread vulnerabilities.Comment: 15 page

Invariant deformation theory of affine schemes with reductive group action

We develop an invariant deformation theory, in a form accessible to practice, for affine schemes $W$ equipped with an action of a reductive algebraic group $G$. Given the defining equations of a $G$-invariant subscheme $X \subset W$, we device an algorithm to compute the universal deformation of $X$ in terms of generators and relations up to a given order. In many situations, our algorithm even computes an algebraization of the universal deformation. As an application, we determine new families of examples of the invariant Hilbert scheme of Alexeev and Brion, where $G$ is a classical group acting on a classical representation, and describe their singularities.Comment: 43 pages, final version, to appear in J. Pure Appl. Algebr

On the symplectic eightfold associated to a Pfaffian cubic fourfold

We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic Pfaffian cubic Y a birational map Z ---> Hilb^4(X), where X is the K3 surface associated to Y by Beauville and Donagi. We interpret Z as a moduli space of complexes on X and observe that at some point of Z, hence on a Zariski open subset, the complex is just the ideal sheaf of four points.Comment: 9 pages. Minor changes; to appear in Crelle as an appendix to 1305.017