The following conjecture may have never been explicitly stated, but seems to have been floating around: If the vertex set of a graph with maximal degree Δ is partitioned into sets Vi of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each Vi at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR's). En route, we give a survey of some recent developments in the theory of ISR's. Independent systems of representativesConsider a graph G and a partition V 1 ,V 2 ,... ,V m of its vertex set. A choice of one vertex from each set V i is called an independent system of representatives (ISR) if the selected vertices are non-adjacent in G. An ISR for a partial family of V 1 ,V 2 ,... ,V m is called a partial ISR. When wishing to stress that an ISR is not properly partial, that is, it is for the entire family V i ,i ≤ m, we sometimes call it a total ISR. The requirement that the sets V i are disjoint is made here only for sake of convenience. The general case, in which the sets may intersect, can be reduced to the case of disjoint sets Mathematics Subject Classification (2000): 05C15

The first result in this direction is due to Matsusaka [6] who gave the following criterion: Let X be an abelian variety of dimension n, BcX a divisor giving a principal polarization, A c X an effective l-cycle homologous to B"-1/(n -1)!; then X is the Jacobian J(A), and, up to a translation, A is canonically embedded in X, and B is the canonical theta-divisor on J(A) (we will abbreviate this conclusion by saying that (X, B, A) is a Jacobian triple). Here we shall prove the following refinement of Matsusaka's criterion: if B c X is any ample divisor, A c X an effective 1-cycle generating X such that the intersection number A. B = n, the smallest possible value, then (X, B, A) is a Jacobian triple.A partial extension of Matsusaka's theorem was obtained by Barton and Clemens [1] who proved, in dimension four, that the locus of principally polarized abelian varieties (X, O) carrying a subvariety homologous to 02/2 is a proper subset of the moduli space. Here we generalize this result by proving that the locus of (X,O) of dimension n carrying subvarieties A, B homologous, respectively, to Od/d! and O" e/(n-d)! contains the Jacobians as an irreducible component. In fact, we prove more precisely that any deformation of a triple (J(C), W,_ e, We), where C is nonhyperelliptic, must be induced by a deformation of C.Our strongest results are in dimension four, where we prove that W 2 subvarieties in Jacobians are characterized among surfaces in abelian fourfolds by having the smallest possible self-intersection number, without being "degenerate". We then apply this result to the Schottky problem, explicitly characterizing Jacobians among principally polarized abelian fourfolds (X,O), essen-QQ~Q-QQ I Q/R 1 IQQ~/QA~Q/~Q~ ?Q

The main purpose of this paper is to prove the unirationality of the moduli space JC/g of curves of genus g for g= 11, 13, over an arbitrary, not necessarily algebraically closed, field; in addition, we extend to an arbitrary field, and provide another proof for Sernesi's theorem [10] that ~12 is unirational over C. In down-to-earth terms, we construct families of curves of genus g=ll, 12, 13, each depending on free parameters and containing 'almost' every birational equivalence class of curves of genus g. In fact, our curves can be described as dependency loci of certain matrices of linear and quadratic forms on p3.To put our results in perspective, we note that the unirationality of ~g was "known" classically for g< 10, at least over C; a rigorous proof of these results was given by Arbarello and Sernesi [2] (for g<8, see also [4]). The unirationality of ~//g12 was recently proved by Sernesi [10]. For genus 1l, S. Mori and S. Mukai have recently proved [-8] that ~'11 is uniruled, i.e., is "filled up" by some family of unirational varieties. On the other hand, Harris and Mumford [5] have proved that for all odd g>23, JC/g is not uniruled, and have asserted an analogous result for all sufficiently large even g.Our method of proof is to show, for each g=ll, 12, 13, the existence of a family of curves Y of genus g and "minimal" degree (i.e., 12, 12, 13 respectively) in p3 having general moduli and a number of other good properties. These properties allow us to represent our curve as the dependency locus of some sections of a vector bundle on p3 which, in turn, can be represented as the cohomology bundle of a particular type of monad. Using one of the good properties of Y, we conclude easily that the space of our monads is rational, and this yields our result.To prove the existence of the aforementioned families, our working method will be to construct certain singular, reducible curves with good properties, and then to smooth them. In this we will be making essential use of recent methods and results of Sernesi [-11], especially his operation of attaching 4-secant conics to a given curve, and his results showing the injectivity of the #0-mapping associated to (_9(1) for the resulting composite curve, and (hence) its embedded deformability to curves with general moduli.

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A second-order invariant of C. Voisin gives a powerful method for bounding from below the geometric genus of a k-dimensional subvariety of a degree-d hypersurface in complex projective n-space. This work uses the Voisin method to establish a general bound, which lies behind recent results of G. Pacienza and Z. Ran.