Incorporated in this 2003 volume are the first two books in Mukai's series on moduli theory. The notion of a moduli space is central to geometry. However, its influence is not confined there; for example, the theory of moduli spaces is a crucial ingredient in the proof of Fermat's last theorem. Researchers and graduate students working in areas ranging from Donaldson or Seiberg-Witten invariants to more concrete problems such as vector bundles on curves will find this to be a valuable resource. Amongst other things this volume includes an improved presentation of the classical foundations of invarant theory that, in addition to geometers, would be useful to those studying representation theory. This translation gives an accurate account of Mukai's influential Japanese texts.

The anticancer peptide PNC-27, which contains an HDM-2-binding domain corresponding to residues 12-26 of p53 and a transmembrane-penetrating domain, has been found to kill cancer cells (but not normal cells) by inducing membranolysis. We find that our previously determined 3D structure of the p53 residues of PNC-27 is directly superimposable on the structure for the same residues bound to HDM-2, suggesting that the peptide may target HDM-2 in the membranes of cancer cells. We now find significant levels of HDM-2 in the membranes of a variety of cancer cells but not in the membranes of several untransformed cell lines. In colocalization experiments, we find that PNC-27 binds to cell membrane-bound HDM-2. We further transfected a plasmid expressing full-length HDM-2 with a membrane-localization signal into untransformed MCF-10-2A cells not susceptible to PNC-27 and found that these cells expressing full-length HDM-2 on their cell surface became susceptible to PNC-27. We conclude that PNC-27 targets HDM-2 in the membranes of cancer cells, allowing it to induce membranolysis of these cells selectively.

The point of this note is to make an observation concerning the variety M(E) parametrizing line subbundles of maximal degree in a generic stable vector bundle E over an algebraic curve C. M(E) is smooth and projective and its dimension is known in terms of the rank and degree of E and the genus of C (see Section 1). Our observation (Theorem 3·1) is that it has exactly the Chern numbers of an étale cover of the symmetric product SδC where δ = dim M(E).This suggests looking for a natural map M(E) → SδC; however, it is not clear what such a map should be. Indeed, we exhibit an example in which M(E) is connected and deforms non-trivially with E, while there are only finitely many isomorphism classes of étale cover of the symmetric product. This shows that for a general deformation in the family M(E) cannot be such a cover (see Section 4).One may conjecture that M(E) is always connected. This would follow from ampleness of a certain Picard-type bundle on the Jacobian and there seems to be some evidence for expecting this, though we do not pursue this question here.Note that by forgetting the inclusion of a maximal line subbundle in E we get a natural map from M(E) to the Jacobian whose image W(E) is analogous to the classical (Brill–Noether) varieties of special line bundles. (In this sense M(E) is precisely a generalization of the symmetric products of C.) In Section 2 we give some results on W(E) which generalise standard Brill–Noether properties. These are due largely to Laumon, to whom the author is grateful for the reference [9].

Abstract. We explore some of the interplay between Brill-Noether subvarieties of the moduli space SU C (2, K) of rank 2 bundles with canonical determinant on a smooth projective curve and 2θ-divisors, via the inclusion of the moduli space into |2Θ|, singular along the Kummer variety. In particular we show that the moduli space contains all the trisecants of the Kummer and deduce that there are quadrisecant lines only if the curve is hyperelliptic; we show that for generic curves of genus < 6, though no higher, bundles with > 2 sections are cut out by Γ 00 ; and that for genus 4 this locus is precisely the Donagi-Izadi nodal cubic threefold associated to the curve.Let SU C (2, L) denote the projective moduli variety of semistable rank 2 vector bundles with determinant L ∈ Pic(C) on a smooth curve C of genus g > 2; and suppose that deg L is even. It is well-known that, on the one hand, the singular locus of SU C (2, L) is isomorphic to the Kummer variety of the Jacobian; and, on the other hand, that when C is nonhyperelliptic SU C (2, O) has an injective morphism into the linear series |2Θ| on the Jacobian J g−1 C which restricts to the Kummer embedding a → Θ a + Θ −a on the singular locus. Dually SU C (2, K) injects into the linear series |L| on J 0 C , where L = O(2Θ κ ) for any theta characteristic κ, and again this map restricts to the Kummer map J g−1 C → |2Θ| ∨ = |L| on the singular locus. This map to projective space (the two cases are of course isomorphic) comes from the complete series on the ample generator of the Picard group, and (at least for a generic curve) is an embedding of the moduli space. Moreover, its image contains much of the geometry studied in connection with the Schottky problem; notably the configuration of Prym-Kummer varieties.In this paper we explore a little of the interplay, via this embedding, between the geometry of vector bundles and the geometry of 2θ-divisors. On the vector bundle side we are principally concerned with the Brill-Noether loci W r ⊂ SU C (2, K) defined by the condition h 0 (E) > r on stable bundles E. These are analogous to the very classical varieties W r g−1 ⊂ J g−1 C . Unlike the line bundle theory, however, general results-connectedness, dimension, smoothness and so on-are not known for the varieties W r (see [6]). On the 2θ side we shall consider the Fay trisecants of the 2θ-embedded Kummer variety, and the subseries PΓ 00 ⊂ |L| consisting of divisors having multiplicity ≥ 4 at the origin. This subseries is known to be important in the study of principally polarised abelian varieties (ppav's) [10]: in the Jacobian of a curve its base locus is the surface C − C ⊂ J whereas for a ppav which is not a Jacobian it is conjectured that the origin is the only base point (but see [3]).The organisation and main results of the paper are as follows. In the first two sections we study two families of lines on SU C (2, K) ⊂ |L| (or equivalently SU C (2, O) ⊂ |2Θ|), each of dimension 3g − 2. These are the Hecke lines, coming from vector bundles of odd degree, on the one h...