Prevention of ECCD triggering explosive bursts in reversed magnetic shear tokamak plasmas for disruption avoidance

The explosive burst excited by neoclassical tearing mode (NTM) is one of the possible candidates of disruptive terminations in reversed magnetic shear (RMS) tokamak plasmas. For the purpose of disruption avoidance, numerical investigations have been implemented on the prevention of explosive burst triggered by the ill-advised application of electron cyclotron current drive (ECCD) in RMS configuration. Under the situation of controlling NTMs by ECCD in RMS tokamak plasmas, a threshold in EC driven current has been found. Below the threshold, not only are the NTM islands not effectively suppressed, but also a deleterious explosive burst could be triggered, which might contribute to the major disruption of tokamak plasmas. In order to prevent this ECCD triggering explosive burst, three control strategies have been attempted in this work and two of them have been recognized to be effective. One is to apply differential poloidal plasma rotation in the proximity of outer rational surface during the ECCD control process; The other is to apply two ECCDs to control NTM islands on both rational surfaces at the same time. In the former strategy, the threshold is diminished due to the modification of classical TM index. In the latter strategy, the prevention is accomplished as a consequence of the reduction of the coupling strength between the two rational surfaces via the stabilization of inner islands. Moreover, the physical mechanism behind the excitation of the explosive burst and the control processes by different control strategies have all been discussed in detail.

Pressure driven long wavelength MHD instabilities in an axisymmetric toroidal resistive plasma

A general set of equations that govern global resistive interchange, resistive internal kink and resistive infernal modes in a toroidal axisymmetric equilibrium are systematically derived in detail. Tractable equations are developed such that resistive effects on the fundamental rational surface can be treated together with resistive effects on the rational surfaces of the sidebands. Resistivity introduces coupling of pressure driven toroidal instabilities with ion acoustic waves, while compression introduces flute-like flows and damping of instabilities, enhanced by toroidal effects. It is shown under which equilibrium conditions global interchange, internal kink modes or infernal modes occur. The m = 1 internal kink is derived for the first time from higher order infernal mode equations, and new resistive infernal modes resonant at the q = 1 surface are reduced analytically. Of particular interest are the competing effects of resistive corrections on the rational surfaces of the fundamental harmonic and on the sidebands, which in this paper is investigated for standard profiles developed for the m = 1 internal kink problem.

Unbounded negativity on rational surfaces in positive characteristic

We give explicit blowups of the projective plane in positive characteristic that contain smooth rational curves of arbitrarily negative self-intersection, showing that the Bounded Negativity Conjecture fails even for rational surfaces in positive characteristic. As a consequence, we show that any surface in positive characteristic admits a birational model failing the Bounded Negativity Conjecture.

Moduli space of J-holomorphic subvarieties

We study the moduli space of J-holomorphic subvarieties in a 4-dimensional symplectic manifold. For an arbitrary tamed almost complex structure, we show that the moduli space of a sphere class is formed by a family of linear system structures as in algebraic geometry. Among the applications, we show various uniqueness results of J-holomorphic subvarieties, e.g. for the fiber and exceptional classes in irrational ruled surfaces. On the other hand, non-uniqueness and other exotic phenomena of subvarieties in complex rational surfaces are explored. In particular, connected subvarieties in an exceptional class with higher genus components are constructed. The moduli space of tori is also discussed, and leads to an extension of the elliptic curve theory.

K3 carpets on minimal rational surfaces and their smoothings

In this paper, we study K3 double structures on minimal rational surfaces [Formula: see text]. The results show there are infinitely many non-split abstract K3 double structures on [Formula: see text] parametrized by [Formula: see text], countably many of which are projective. For [Formula: see text] there exists a unique non-split abstract K3 double structure which is non-projective (see [J.-M. Drézet, Primitive multiple schemes, preprint (2020), arXiv:2004.04921 , to appear in Eur. J. Math.]). We show that all projective K3 carpets can be smoothed to a smooth K3 surface. One of the byproducts of the proof shows that unless [Formula: see text] is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on [Formula: see text]. Moreover, we show any embedded projective K3 carpet on [Formula: see text] with [Formula: see text] arises as a flat limit of embeddings degenerating to 2:1 morphism. The rest do not, but we still prove the smoothing result. We further show that the Hilbert points corresponding to the projective K3 carpets supported on [Formula: see text], embedded by a complete linear series are smooth points if and only if [Formula: see text]. In contrast, Hilbert points corresponding to projective (split) K3 carpets supported on [Formula: see text] and embedded by a complete linear series are always smooth. The results in [P. Bangere, F. J. Gallego and M. González, Deformations of hyperelliptic and generalized hyperelliptic polarized varieties, preprint (2020), arXiv:2005.00342 ] show that there are no higher dimensional analogues of the results in this paper.

Rational conic fibrations of sectional genus two

Polarized rational surfaces (X, L) of sectional genus two ruled in conics are studied. When they are not minimal, they are described as the blow-up of 𝔽1 at some points lying on distinct fibers. Ampleness and very ampleness of L are studied in terms of their location. When L is very ample and there is a line contained in X and transverse to the fibers, the conic fibrations (X, L) are classified and a related property concerned with the inflectional locus is discussed.

Covering Rational Surfaces with Rational Parametrization Images

Let S be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps f,g,h:A2⇢S⊂Pn such that the union of the three images covers S. As a consequence, we present a second algorithm that generates two rational maps f,g˜:A2⇢S, such that the union of its images covers the affine surface S∩An. In the affine case, the number of rational maps involved in the cover is in general optimal.