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

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

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

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

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

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

The Anti-Sigma Factor MucA of Pseudomonas aeruginosa: Dramatic Differences of a mucA22 vs. a ΔmucA Mutant in Anaerobic Acidified Nitrite Sensitivity of Planktonic and Biofilm Bacteria in vitro and During Chronic Murine Lung Infection

Mucoid mucA22 Pseudomonas aeruginosa (PA) is an opportunistic lung pathogen of cystic fibrosis (CF) and chronic obstructive pulmonary disease (COPD) patients that is highly sensitive to acidified nitrite (A-NO2-). In this study, we first screened PA mutant strains for sensitivity or resistance to 20 mM A-NO2- under anaerobic conditions that represent the chronic stages of the aforementioned diseases. Mutants found to be sensitive to A-NO2- included PA0964 (pmpR, PQS biosynthesis), PA4455 (probable ABC transporter permease), katA (major catalase, KatA) and rhlR (quorum sensing regulator). In contrast, mutants lacking PA0450 (a putative phosphate transporter) and PA1505 (moaA2) were A-NO2- resistant. However, we were puzzled when we discovered that mucA22 mutant bacteria, a frequently isolated mucA allele in CF and to a lesser extent COPD, were more sensitive to A-NO2- than a truncated ΔmucA deletion (Δ157–194) mutant in planktonic and biofilm culture, as well as during a chronic murine lung infection. Subsequent transcriptional profiling of anaerobic, A-NO2--treated bacteria revealed restoration of near wild-type transcript levels of protective NO2- and nitric oxide (NO) reductase (nirS and norCB, respectively) in the ΔmucA mutant in contrast to extremely low levels in the A-NO2--sensitive mucA22 mutant. Proteins that were S-nitrosylated by NO derived from A-NO2- reduction in the sensitive mucA22 strain were those involved in anaerobic respiration (NirQ, NirS), pyruvate fermentation (UspK), global gene regulation (Vfr), the TCA cycle (succinate dehydrogenase, SdhB) and several double mutants were even more sensitive to A-NO2-. Bioinformatic-based data point to future studies designed to elucidate potential cellular binding partners for MucA and MucA22. Given that A-NO2- is a potentially viable treatment strategy to combat PA and other infections, this study offers novel developments as to how clinicians might better treat problematic PA infections in COPD and CF airway diseases

An extremal effective survey about extremal effective cycles in moduli spaces of curves

We survey recent developments and open problems about extremal effective divisors and higher codimension cycles in moduli spaces of curves.Comment: Submitted to the Proceedings of the Abel Symposium 2017. Comments are welcom