Limit linear series: Basic theory

In this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves, curves of compact type. The technically much simpler special case of 1-dimensional series was developed by Beauville [2], Knudsen [21–23], Harris and Mumford [17], in the guise of “admissible covers”. It has proved very useful for studying the Moduli space of curves (the above papers and Harris [16]) and the simplest sorts of Weierstrass points (Diaz [4]). With our extended tools we are able to prove, for example, that:

1. The Moduli spaceM _g of curves of genusg has general type forg≧24, and has Kodaira dimension ≧1 forg=23, extending and simplifying the work of Harris and Mumford [17] and Harris [16].
2. Given a Weierstrass semigroup Γ of genusg and weightw≦g/2 (and in a somewhat more general case) there exists at least one component of the subvariety ofM _g of curves possessing a Weierstrass point of semigroup Γ which has the “expected” dimension 3g-2?w (and in particular, this set is not empty).
3. The fundamental group of the space of smooth genusg curves having distinct “ordinary” Weierstrass points acts on the Weierstrass points by monodromy as the full symmetric group.
4. Ifr andd are chosen so that $$\rho : = g - (r + 1)(g - d + r) = 0,$$ then the general curve of genusg has a certain finite number ofg _d ^r’ s [15, 20]. We show that the family of all these, allowing the curve to vary among general curves, is irreducible, so that the monodromy of this family acts transitively. If4=1, we show further that the monodromy acts as the full symmetric group.
5. Ifr andd are chosen so that $$\rho = - 1,$$ then the subvariety ofM _g consisting of curves posessing ag _d ^r has exactly one irreducible component of codimension 1.
6. For anyr, g, d such that ρ≦0, the subvariety ofM _g consisting of curves possessing ag _d ^r has at least one irreducible component of codimension—ρ so long as $$\rho \geqq \left\{ \begin{gathered} - g + r + 3 (r odd) \hfill \\ - \frac{r}{{r + 2}}g + r + 3 (r even). \hfill \\ \end{gathered} \right.$$

In this paper we present the basic theory of “limit linear series” necessary for proving these results. The results themselves will be taken up in our forthcoming papers [8-12]. Simpler applications, not requiring the tools developed in this paper but perhaps clarified by them, have already been given in our papers [5-7].     

Size-controlled synthesis and characterization of thiol-stabilized gold nanoparticles

Size-controlled synthesis of nanoparticles of less than a few nanometers in size is a challenge due to the spatial resolution limit of most scattering and imaging techniques used for their structural characterization. We present the self-consistent analysis of the extended x-ray absorption fine-structure (EXAFS) spectroscopy data of ligand-stabilized metal nanoclusters. Our method employs the coordination number truncation and the surface-tension models in order to measure the average diameter and analyze the structure of the nanoparticles. EXAFS analysis was performed on the two series of dodecanethiol-stabilized gold nanoparticles prepared by one-phase and two-phase syntheses where the only control parameter was the gold/thiol ratio xi, varied between 6:1 and 1:6. The two-phase synthesis resulted in the smaller particles whose size decreased monotonically and stabilized at 16 A when xi was lowered below 1:1. This behavior is consistent with the theoretically predicted thermodynamic limit obtained previously in the framework of the spherical drop model of Au nanoparticles.     

Chain-Cleavage and hydrolysis of activated polyethylene glycol derivatives: Evidence for competitive processes

Studies have been carried out with the tosylate of the monomethyl ether of polyethylene glycol (MeO–PEG–OTs) and with low molecular weight models to assess whether the neighboring oxygen at position 3 or 6 provides the driving force for hydrolytic cleavage of these activated derivatives. Our results reveal that MeO–PEG–OTs undergoes hydrolysis by competitive pathways. Water directly displaces the tosylate group to give the original PEG alcohol and the oxygen at position 6 nucleophilically displaces the tosylate group to give a cyclic oxonium ion as an intermediate. This intermediate can react by three pathways. First, it can lead to the production of the original PEG alcohol by attack of water on a ring carbon; second, dioxane and a lower molecular weight PEG alcohol is produced by water attack at the nonring carbon next to the charged oxygen; and third dioxane can be displaced by the oxygen atom at position 6 in the chain.     

An alternative computational strategy for direct space hartree–fock calculations on extended model chains

Based on previous analyses of slowly convergent exchange lattice sums entering the configuration space restricted Hartree–Fock–Roothaan scheme for chain systems, an alternative computational strategy is developed. Within the present formalism, the traditionally used finite Fourier transform of k-dependent LCAO density matrices are by-passed and an advantageous computational organization is obtained.     

Reactions of stabilizer compounds. III. Oxidative reactions of some 2′-hydroxy-2-phenylbenzotriazole light stabilizers in the AIBN-initiated autoxidation of cumene

Oxidative decay of number of commercially important 2′-hydroxy-2-phenylbenzotriazole light stabilizers during the AIBN-initiated autoxidation of cumene at 65°C is described. The reactivity of the hydroxyphenylbenzotriazoles studied increased in the order 2-(2′-hydroxy-5′-methyl)phenylbenzotriazole (Ia) < 2-(2′-hydroxy-3′,5′ -di-tert-pentyl)phenylbenzotriazole (Ie) < 5-chloro-2-(3′,5′-di-tert-butyl-2′-hydroxy)phenylbenzotriazole (Ic) < 5-chloro-2-(3′-tert-butyl-2′-hydroxy-5′-methyl)phenylbenzotriazole (Ib). The major product from the reaction of Ib was identified as the cumylperoxycyclohexa-2,5-dienone (III). The possible occurrence of these reactions during the degradation of stabilized polymers is discussed.