A computational approach to Lüroth quartics

A plane quartic curve is called Lüroth if it contains the ten vertices of a complete pentalateral. White and Miller constructed in 1909 a covariant quartic fourfold, associated to any plane quartic. We review their construction and we show how it gives a computational tool to detect if a plane quartic is Lüroth. As a byproduct, we show that the 28 bitangents of a general plane quartic correspond to 28 singular points of the associated White–Miller quartic fourfold.     

A counterexample to Valette’s conjecture

We disprove a well-known conjecture of D. Vallete (1978), which states that every d-dimensional self-affine convex body is a direct product of a polytope with a convex body of lower dimension. It is shown that there are counterexamples for dimension d = 4. Additional assumptions under which the conjecture is true are discussed.     

Baumgartnerʼs conjecture and bounded forcing axioms

We study the spectrum of forcing notions between the iterations of σ-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of α-proper forcings for indecomposable countable ordinals α, the Axiom A forcings and forcings completely embeddable into an iteration of a σ-closed followed by a ccc forcing. For the latter class, we present an equivalent characterization in terms of Baumgartner?s Axiom A. This resolves a conjecture of Baumgartner from the 1980s.     

Perverse equivalences and Brouéʼs conjecture

We give a new approach to the construction of derived equivalences between blocks of finite groups, based on perverse equivalences, in the setting of Broué?s abelian defect group conjecture. We provide in particular local and global perversity data describing the principal blocks and the derived equivalences for a number of finite simple groups with Sylow subgroups elementary abelian of order 9. We also examine extensions to automorphism groups in a general setting.     

On Serreʼs conjecture over imaginary quadratic fields

We study an analog of Serre?s conjecture over imaginary quadratic fields. In particular, we ask whether the weight recipe of Buzzard, Diamond and Jarvis will hold in this setting. Using a program written by the author, we provide computational evidence that this is in fact the case. In order to justify the method used in the program, we prove that a modular symbols method will work for arbitrary weights over imaginary quadratic fields.     

Viehwegʼs hyperbolicity conjecture is true over compact bases

We prove Viehweg?s hyperbolicity conjecture over compact bases and over bases with non-uniruled compactification. The most general case of the conjecture states that the base space of a maximal variation family of smooth projective manifolds with semi-ample canonical sheaf is of log-general type.     

Counterexamples to Mercat’s conjecture

In this paper, we provide counterexamples to Mercat’s conjecture on vector bundles on algebraic curves. For any \({n \geq 4}\), we provide examples of curves lying on K3 surfaces and vector bundles on those curves which invalidate Mercat’s conjecture in rank n.     

A new approach to Sheppard’s corrections

A very simple closed-form formula for Sheppard’s corrections is recovered by means of the classical umbral calculus. Using this symbolic method, a more general closed-form formula for discrete parent distributions is provided and the generalization to the multivariate case turns out to be straightforward. All these new formulas are particularly suited to be implemented in any symbolic package.     

A Galois-theoretic approach to Kanev’s correspondence

Let G be a finite group, an absolutely irreducible -module and w a weight of . To any Galois covering with group G we associate two correspondences, the Schur and the Kanev correspondence. We work out their relation and compute their invariants. Using this, we give some new examples of Prym–Tyurin varieties. This work was supported by FONDECYT No. 11060468 and No. 1060742.     

A relaxation approach to hencky’s plasticity

Givenμ, κ, c>0, we consider the functional