Let C be a general element in the locus of curves in Mg lying on some K3 surface, where g is congruent to 3 mod 4 and greater than or equal to 15. Following Mukai's ideas, we show how to reconstruct the K3 surface as a Fourier-Mukai transform of a Brill-Noether locus of rank two vector bundles on C.

Let C be a Brill-Noether-Petri curve of genus g 12. We prove that C lies on a polarised K3 surface, or on a limit thereof, if and only if the Gauss-Wahl map for C is not surjective. The proof is obtained by studying the validity of two conjectures by J. Wahl. Let I C be the ideal sheaf of a non-hyperelliptic, genus g, canonical curve. The first conjecture states that if g 8 and if the Clifford index of C is greater than 2, then H 1 (P g−1 , I 2 C (k)) = 0 for k 3. We prove this conjecture for g 11. The second conjecture states that a Brill-Noether-Petri curve of genus g 12 is extendable if and only if C lies on a K3 surface. As observed in the introduction, the correct version of this conjecture should admit limits of polarised K3 surfaces in its statement. This is what we prove in the present work.

IntroductionThe moduli space M g,n of smooth n-pointed curves of genus g, and its projective closure, the Deligne-Mumford compactification M g,n , is a classical object of study that reflects many of the properties of families of pointed curves. As a matter of fact, the study of its biregular geometry is of interest in itself and has become a central theme in various areas of mathematics.Already for small n, the moduli spaces M 0,n are quite intricate objects deeply rooted in classical algebraic geometry. Under this perspective, Kapranov showed in [Ka] that M 0,n is identified with the closure of the subscheme of the Hilbert scheme parametrizing rational normal curves passing through n points in linearly general position in P n−2 . Via this identification, given n − 1 points in linearly general position in P n−3 , M 0,n is isomorphic to an iterated blow-up of P n−3 at the strict transforms of all the linear spaces spanned by subsets of the points in order of increasing dimension. In a natural way, then, base point free linear systems on M 0,n are identified with linear systems on P n−3 whose base locus is quite special and supported on so-called vital spaces, i.e. spans of subsets of the given points. Another feature of this picture is that all these vital spaces correspond to divisors in M 0,n which have a modular interpretation as products of M 0,r for r < n. In this interpretation, the modular forgetful maps φ I : M 0,n → M 0,n−|I| , which forget points indexed by I ⊂ {1, . . . , n}, correspond, up to standard Cremona transformations, to linear projections from vital spaces. The aim of this paper is to study automorphisms of M 0,n with the aid of Kapranov's beautiful description.It is expected that the only possible biregular automorphisms of M 0,n are the one associated to a permutation of the markings. Any such morphism has to permute the forgetful maps onto M 0,n−1 as well. This induces, on P n−3 , special birational maps that switch lines through n − 1 points in general position. On the other hand if we were able to prove that any automorphism has to permute forgetful maps this should lead to a proof that every automorphism is a permutation. Our main tool to classify Aut(M 0,n ) is therefore the following Theorem.Theorem 1. Let f : M 0,n → M 0,r1 × . . . × M 0,r h be a dominant morphism with connected fibers. Then f is a forgetful map.

A kinetic theory for the equilibrium of an unmagnetized plasma consisting of electrons and ions surrounding a spherical body emitting electrons (due to thermionic emission, photoemission, or secondary emission) is presented. The theory is valid for positively charged bodies, neglects collisions of the plasma particles, and is formulated for profiles of the shielding potential presenting an attractive well. Particle-in-cell simulations are shown to be in good agreement with the theory. An approximated criterion is derived to determine the presence of the potential well.

The purpose of this paper is two-fold. The first is to give a tutorial introduction to the so called Sarkisov program, a 3-dimensional generalization of Castelnuovo-Nöther Theorem "untwisting" birational maps between Mori fiber spaces, which was recently established by Corti[4]. We should emphasize that though the general features were understood (cf.Matsuki[17]) after Reid[29] explained the original ideas of V.G. Sarkisov in a substantially laundered form, it is only after Corti[4] that we are beginning to understand the details of the mechanism. Here we will present a flowchart to visualize how the Sarkisov program works and also slightly simplify the proof of termination after the ingenious argument of Corti[4]: we prove there is no infinite loop in the program just observing that the Sarkisov degree decreases strictly after each untwisting and it cannot decrease infinitely many times using the boundedness of Q-Fano d-folds d ≤ 3 together with the ascending chain condition S 3 (Local) of Alexeev[1] (cf.Shokurov[31]Kollár et al[16]). Our argument also makes it explicit that the Sarkisov program holds in arbitrary dimension n once we have Log MMP in dimension n, boundedness of Q-Fano d-folds for d ≤ n and S n (Local). The second is an attempt to give a logarithmic generalization following the philosophy of Iitaka, based upon the Log MMP (established in dimension 3 by Shokurov[31]Kawamata[9] (cf.Kollár et al[16])). The key is to understand the meaning of the genuine Sarkisov program and set up the natural and right generalization. The genuine Sarkisov program untwists any birational map between two