G-torsors and universal torsors over nonsplit del Pezzo surfaces

Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of $S_L$ in Pic $S_L$, or a form of it containing the Neron-Severi torus. Let $mathcal{G}$ be the G-torsor over $S_L$ obtained by extension of structure group from a universal torsor $mathcal{T}$ over $S_L$. We prove that $mathcal{G}$ does not descend to S unless $mathcal{T}$ does. This is in contrast to a result of Friedman and Morgan that such $mathcal{G}$ always descend to singular del Pezzo surfaces over $mathbb{C}$ from their desingularizations.

The Peregrine breather on the zero-background limit as the two-soliton degenerate solution: An experimental study

Solitons are coherent structures that describe the nonlinear evolution of wave localizations in hydrodynamics, optics, plasma and Bose-Einstein condensates. While the Peregrine breather is known to amplify a single localized perturbation of a carrier wave of finite amplitude by a factor of three, there is a counterpart solution on zero background known as the degenerate two-soliton which also leads to high amplitude maxima. In this study, we report several observations of such multi-soliton with doubly-localized peaks in a water wave flume. The data collected in this experiment confirm the distinctive attainment of wave amplification by a factor of two in good agreement with the dynamics of the nonlinear Schrodinger equation solution. Advanced numerical simulations solving the problem of nonlinear free water surface boundary conditions of an ideal fluid quantify the physical limitations of the degenerate two-soliton in hydrodynamics.

A birational Torelli theorem with a Brauer class

Let $text{M}_C( 2, mathcal{O}_C) cong mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $mathbb{C}$. Let $beta_C$ denote the natural Brauer class over the stable locus. We prove that if $f^*( beta_{C}) = beta_C$ for some birational map $f$ from $text{M}_C( 2, mathcal{O}_C)$ to $text{M}_{C}( 2, mathcal{O}_{C})$, then the Jacobians of $C$ and of $C$ are isomorphic as abelian varieties. If moreover these Jacobians do not admit real multiplication, then the curves $C$ and $C$ are isomorphic. Similar statements hold for Kummer surfaces in $mathbb{P}^3$ and for quadratic line complexes.

Energy harvesting below the onset of flutter

This work demonstrates preliminary results on energy harvesting from a linearly stable flutter-type system with circulatory friction forces. Harmonic external forcing is applied to study the energy flow in the steady sliding configuration. In certain parameter ranges negative excitation work is observed where the external forcing allows to pull part of the friction energy out of the system and thus makes energy harvesting possible. Studies reveal that this behavior is largely independent of the flutter point and thus that it is primarily controlled by the excitation. Contrary to existing energy harvesting approaches for such systems, this approach uses external forcing in the linearly stable regime of the oscillator which allows to control vibrations and harvest energy on demand.

Directional Soliton and Breather Beams

Solitons and breathers are nonlinear modes that exist in a wide range of physical systems. They are fundamental solutions of a number of nonlinear wave evolution equations, including the uni-directional nonlinear Schrodinger equation (NLSE). We report the observation of slanted solitons and breathers propagating at an angle with respect to the direction of propagation of the wave field. As the coherence is diagonal, the scale in the crest direction becomes finite, consequently, a beam dynamics forms. Spatio-temporal measurements of the water surface elevation are obtained by stereo-reconstructing the positions of the floating markers placed on a regular lattice and recorded with two synchronized high-speed cameras. Experimental results, based on the predictions obtained from the (2D+1) hyperbolic NLSE equation, are in excellent agreement with the theory. Our study proves the existence of such unique and coherent wave packets and has serious implications for practical applications in optical sciences and physical oceanography. Moreover, unstable wave fields in this geometry may explain the formation of directional large amplitude rogue waves with a finite crest length within a wide range of nonlinear dispersive media, such as Bose-Einstein condensates, plasma, hydrodynamics and optics.

On the rationality problem for forms of moduli spaces of stable marked curves of positive genus

Let $M_{g, n}$ (respectively, $overline{M_{g, n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not $M_{g, n}$ (or equivalently, $overline{M_{g, n}}$) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that $M_{g, n}$ is not unirational for any $n geqslant 0$ if $g geqslant 22$. Moreover, P. Belorousski and A. Logan showed that $M_{g, n}$ is unirational for only finitely many pairs $(g, n)$ with $g geqslant 1$. Finding the precise range of pairs $(g, n)$, where $M_{g, n}$ is rational, stably rational or unirational, is a problem of ongoing interest. In this paper we address the rationality problem for twisted forms of $overline{M_{g, n}}$ defined over an arbitrary field $F$ of characteristic $ eq 2$. We show that all $F$-forms of $overline{M_{g, n}}$ are stably rational for $g = 1$ and $3 leqslant n leqslant 4$, $g = 2$ and $2 leqslant n leqslant 3$, $g = 3$ and $1 leqslant n leqslant 14$, $g = 4$ and $1 leqslant n leqslant 9$, $g = 5$ and $1 leqslant n leqslant 12$.

Stability of the Poincare bundle

Let X be an irreducible smooth projective curve, of genus at least two, over an algebraically closed field k. Let $mathcal{M}^d_G$ denote the moduli stack of principal G-bundles over X of fixed topological type $d in pi_1(G)$, where G is any almost simple affine algebraic group over k. We prove that the universal bundle over $X times mathcal{M}^d_G$ is stable with respect to any polarization on $X times mathcal{M}^d_G$. A similar result is proved for the Poincare adjoint bundle over $X times M_G^{d, rs}$, where $M_G^{d, rs}$ is the coarse moduli space of regularly stable principal G-bundles over X of fixed topological type d.

Degeneration of torsors over families of del Pezzo surfaces

Let S be a split family of del Pezzo surfaces over a discrete valuation ring such that the general fiber is smooth and the special fiber has ADE-singularities. Let G be the reductive group given by the root system of these singularities. We construct a G-torsor over S whose restriction to the generic fiber is the extension of structure group of the universal torsor. This extends a construction of Friedman and Morgan for individual singular del Pezzo surfaces. In case of very good residue characteristic, this torsor is unique and infinitesimally rigid.

Torelli theorem for the Deligne--Hitchin moduli space

Fix integers $ggeq 3$ and $rgeq 2$, with $rgeq 3$ if $g=3$. Given a compact connected Riemann surface $X$ of genus $g$, let $MDH(X)$ denote the corresponding $text{SL}(r, {mathbb C})$ Deligne--Hitchin moduli space. We prove that the complex analytic space $MDH(X)$ determines (up to an isomorphism) the unordered pair ${X, overline{X}}$, where $overline{X}$ is the Riemann surface defined by the opposite almost complex structure on $X$.