Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces

We provide explicit descriptions of the generic members of Hassett's divisors $\mathcal C_d$ for relevant $18\leq d\leq 38$ and for $d=44$. In doing so, we prove that $\mathcal C_d$ is unirational for $18\leq d\leq 38,d=44$. As a corollary, we prove that the moduli space $\mathcal N_{d}$ of polarized K3 surfaces of degree $d$ is unirational for $d=14,26,38$. The case $d=26$ is entirely new, while the other two cases have been previously proven by Mukai.Comment: 13 pages, 2 tables. Script for the computer calculations used are provided on the author's websit

On the automorphisms of moduli spaces of curves

In the last years the biregular automorphisms of the Deligne-Mumford's and Hassett's compactifications of the moduli space of n-pointed genus g smooth curves have been extensively studied by A. Bruno and the authors. In this paper we give a survey of these recent results and extend our techniques to some moduli spaces appearing as intermediate steps of the Kapranov's and Keel's realizations of $\bar{M}_{0,n}$, and to the degenerations of Hassett's spaces obtained by allowing zero weights.Comment: 15 pages. The material of version 1 has been reorganized and expanded in this paper and in arXiv:1307.6828 on automorphisms of Hassett's moduli space

Brauer groups and quotient stacks

A natural question is to determine which algebraic stacks are qoutient stacks. In this paper we give some partial answers and relate it to the old question of whether, for a scheme X, the natural map from the Brauer goup (equivalence classes of Azumaya algebras) to the cohomological Brauer group (the torsion subgroup of $H^2(X,{\mathbb G}_m)$ is surjective.Comment: American J. Math, to appear. (Latex2e, 17pp

Log Fano varieties over function fields of curves

Consider a smooth log Fano variety over the function field of a curve. Suppose that the boundary has positive normal bundle. Choose an integral model over the curve. Then integral points are Zariski dense, after removing an explicit finite set of points on the base curve.Comment: 18 page

Gravitational dynamics for all tensorial spacetimes carrying predictive, interpretable and quantizable matter

Only a severely restricted class of tensor fields can provide classical spacetime geometries, namely those that can carry matter field equations that are predictive, interpretable and quantizable. These three conditions on matter translate into three corresponding algebraic conditions on the underlying tensorial geometry, namely to be hyperbolic, time-orientable and energy-distinguishing. Lorentzian metrics, on which general relativity and the standard model of particle physics are built, present just the simplest tensorial spacetime geometry satisfying these conditions. The problem of finding gravitational dynamics---for the general tensorial spacetime geometries satisfying the above minimum requirements---is reformulated in this paper as a system of linear partial differential equations, in the sense that their solutions yield the actions governing the corresponding spacetime geometry. Thus the search for modified gravitational dynamics is reduced to a clear mathematical task.Comment: 47 pages, no figures, minor update

Derived categories of cubic fourfolds

We discuss the structure of the derived category of coherent sheaves on cubic fourfolds of three types: Pfaffian cubics, cubics containing a plane and singular cubics, and discuss its relation to the rationality of these cubics.Comment: 18 page

A characterization of compact complex tori via automorphism groups

We show that a compact Kaehler manifold X is a complex torus if both the continuous part and discrete part of some automorphism group G of X are infinite groups, unless X is bimeromorphic to a non-trivial G-equivariant fibration. Some applications to dynamics are given.Comment: title changed, to appear in Math. An

Geometric invariant theory of syzygies, with applications to moduli spaces

We define syzygy points of projective schemes, and introduce a program of studying their GIT stability. Then we describe two cases where we have managed to make some progress in this program, that of polarized K3 surfaces of odd genus, and of genus six canonical curves. Applications of our results include effectivity statements for divisor classes on the moduli space of odd genus K3 surfaces, and a new construction in the Hassett-Keel program for the moduli space of genus six curves.Comment: v1: 23 pages, submitted to the Proceedings of the Abel Symposium 2017, v2: final version, corrects a sign error and resulting divisor class calculations on the moduli space of K3 surfaces in Section 5, other minor changes, In: Christophersen J., Ranestad K. (eds) Geometry of Moduli. Abelsymposium 2017. Abel Symposia, vol 14. Springer, Cha

Video and computer-based interactive exercises are safe and improve task-specific balance in geriatric and neurological rehabilitation: A randomised trial

© 2015. Question: Does adding video/computer-based interactive exercises to inpatient geriatric and neurological rehabilitation improve mobility outcomes? Is it feasible and safe? Design: Randomised trial. Participants: Fifty-eight rehabilitation inpatients. Intervention: Physiotherapist-prescribed, tailored, video/computer-based interactive exercises for 1 hour on weekdays, mainly involving stepping and weight-shifting exercises. Outcome measures: The primary outcome was the Short Physical Performance Battery (0 to 3) at 2 weeks. Secondary outcomes were: Maximal Balance Range (mm); Step Test (step count); Rivermead Mobility Index (0 to 15); activity levels; Activity Measure for Post Acute Care Basic Mobility (18 to 72) and Daily Activity (15 to 60); Falls Efficacy Scale (10 to 40), ED5D utility score (0 to 1); Reintegration to Normal Living Index (0 to 100); System Usability Scale (0 to 100) and Physical Activity Enjoyment Scale (0 to 126). Safety was determined from adverse events during intervention. Results: At 2 weeks the between-group difference in the primary outcome (0.1, 95% CI -0.2 to 0.3) was not statistically significant. The intervention group performed significantly better than usual care for Maximal Balance Range (38. mm difference after baseline adjustment, 95% CI 6 to 69). Other secondary outcomes were not statistically significant. Fifty-eight (55%) of the eligible patients agreed to participate, 25/29 (86%) completed the intervention and 10 (39%) attended > 70% of sessions, with a mean of 5.6 sessions (SD 3.3) attended and overall average duration of 4.5. hours (SD 3.1). Average scores were 62 (SD 21) for the System Usability Scale and 62 (SD 8) for the Physical Activity Enjoyment Scale. There were no adverse events. Conclusion: The addition of video/computer-based interactive exercises to usual rehabilitation is a safe and feasible way to increase exercise dose, but is not suitable for all. Adding the exercises to usual rehabilitation resulted in task-specific improvements in balance but not overall mobility. Registration: ACTRN12613000610730