the geometry of the moduli space of curves of genus 23 the problem of describing the birational geometry of the moduli space mg of complex curves of genus g has a long history severi already knew in 1915 thatmg is unirational for g 10 cf sev see also ac1 for a modern proof in the same paper severi conjectured thatmg is unirational for all genera g then for a long period this problem seemed intractable mumford writes in mu p 51 whether more mg s g 11 are unirational or not is a very interesting problem but one which looks very hard too especially if g is quite large the breakthrough came in the eighties when eisenbud harris and mumford proved that mg is of general type as soon as g 24 and that the kodaira dimension of m23 is 1 see hm eh3 we note that mg is rational for g 6 see dol for problems concerning the rationality of various moduli spaces severi s proof of the unirationality of mg for small g was based on representing a general curve of genus g as a plane curve of degree d with nodes this is possible when d 2g 3 2 when the number of nodes is small i e d 1 d 2 6 the dominant map from the variety of plane curves of degree d and genus g to mg yields a rational parametrization of the moduli space the two conditions involving d and can be satisfied only when g 10 so severi s argument cannot be extended for other genera however using much more subtle ideas chang ran and sernesi proved the unirationality of mg for g 11 12 13 see cr1 se1 while for g 15 16 they proved that the kodaira dimension is see cr2 4 the remaining cases g 14 and 17 g 23 are still quite mysterious harris and morrison conjectured in hmo that mg is uniruled precisely when g 23 all these facts indicate that m23 is a very interesting transition case our main result is the following

the mercat conjecture for stable rank 2 vector bundles on generic curves we prove the mercat conjecture for rank 2 vector bundles on generic curves of every genus for odd genus we identify the effective divisor in mg where the mercat conjecture fails and compute its slope the clifford index of a smooth curve c defined as the non negative quantity cliff c min deg l 2r l l pic c h c l 2 deg l g 1 stratifies the moduli space mg of curves of genus g with the smallest stratum being that consisting of hyperelliptic curves which have clifford index zero while the definition is obviously inspired by the classical clifford theorem the notion came to real prominence in the works of green lazarsfeld g gl voisin v2 and many others in the context of syzygies of canonical curves for a general curve c of genus g one has cliff c bg 1 2 c the locus of curves c mg with cliff c bg 1 2 c is a subvariety of codimension 1 respectively 2 when g is odd respectively even much studied by harris and mumford hm it has been a long standing problem to find an adequate definition of the clifford index for higher rank vector bundles on curves a higher rank clifford index should not only capture the behavior of the generic curve from the point of view of special higher rank vector bundles but also provide a geometrically meaningful stratification ofmg an interesting definition has been put forward by lange and newstead ln1 for a semistable vector bundle e of rank 2 and slope e on a curve c of genus g one defines its clifford index as cliff e e h c e 2 0 the rank 2 clifford index of c is then defined as the quantity cliff2 c min cliff e e uc 2 d d 2g 2 h c e 4 observe that if l is a line bundle on c with deg l 2h0 c l 2 cliff c that is l computes the clifford index of c then cliff l l cliff c in particular the inequality cliff2 c cliff c holds for every curve c of genus g the main achievement of this paper is the proof of the following result theorem 1 for a general curve c of genus g the following equality holds cliff2 c cliff c g 1 2 equivalently if e is a semistable rank 2 vector bundle on c contributing to cliff2 c then h c e e 2 cliff c mercat me conjectured the equality cliff2 c cliff c for every smooth curve c mg counterexamples to this expectation have been found using noether lefschetz special k3 surfaces in fo1 fo2 and ln2 however these examples are special in moduli and theorem 1 proves mercat s conjecture for generic curves of every genus

on the existence of certain families of curves introduction 25 w 1 generalities on families of curves in ip 28 w the map 0 d 33 w 3 the number of moduli of a family 40 w 4 plane curves existence of families with the expected number of moduli 43 w 5 smoothing certain reducible curves in ip r r 3 47 w 6 curves in ip r 3 existence of families with the expected number of moduli 52 w curves in ip3 a more general existence theorem 55

a sharp nonconvexity bound for partition ranges of vector measures with atoms a sharp upper bound is given for the degree of nonconvexity of the partition range of a finite dimensional vector measure in terms of the maximum one dimensional mass of the atoms of that measure this upper bound improves on a bound of hill and tong 1989 anal stat 17 1325 1334 by an order of magnitude n its proof uses several ideas from graph theory combinatorics and convex geometry applications are given to optimal partitioning and fair division problems

optimal partitioning inequalities for nonatomic probability measures on considere une inegalite qui affine et quantifie un theoreme de decoupe du a urbanik 1955 et dubins et spanier 1961

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 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

self consistent projection operator method for describing the non markovian time evolution of subsystems it is pointed out that the usual projection operator method yields non positive definite reduced density operators for subsystems in order to obtain non negative probabilities a new self consistent projection operator method for treating non markovian behaviour is developed finally the applicability of the method is shown by an example

the dubins spanier optimization problem in fair division theory abstract an optimality criterion for fair division theory introduced by dubins and spanier in 1961 is recalled with the purpose of analyzing its structure in relationship with other optimal solutions pareto and equi optimal partitions a geometric dual approach is also defined with the purpose of characterizing and identifying the dubins spanier optimal solutions

optimal control of quantum systems a projection approach this paper considers the optimal control of quantum systems the controlled quantum systems are described by the probability density matrix based liouville von neumann equation using projection operators the states of the quantum system are decomposed into two sub spaces namely the main state space and the remaining state space since the control energy is limited a solution for optimizing the external control force is proposed in which the main state is brought to the desired main state at a certain target time while the population of the remaining state is simultaneously suppressed in order to diminish its effects on the final population of the main state the optimization problem is formulated by maximizing a general cost functional of states and control force an efficient algorithm is developed to solve the optimization problem finally using the hydrogen fluoride hf molecular population transfer problem as an illustrative example the effectiveness of the proposed scheme for a quantum system initially in a mixed state or in a pure state is investigated through numerical simulations

the gonality of general smooth curves with a prescribed plane nodal model let c be a smooth curve of genus g it is well known that c possesses a linear system g l if q l g 2n g 2 0 moreover if c is general and q g 2 k l and 6 kd k l 2 3 theorem 2 1 in loc cit and for each d this bound on 6 is sharp examples 4 1 and 4 2 in loc cit in this paper we consider the question for a general choice of f since the space parametrizing integral plane nodal curves with prescribed d and 6 is irreducible famous conjecture of severi proved by harris 9 the meaning of general becomes represented by a point of a particular nonempty zariski open subset of that parameter space note that qn x 3 g 0 if and only if 6 d 2 7d 18 2

divisors on general curves and cuspidal rational curves in t roduc t ion 371 0 pre l iminar ies 377 1 a pl i icker formula 378 2 d imens iona l t ransversa l i ty 380 3 d i m e n s i o n of families of r a t iona l curves with fixed degree and ass igned cusps in a project ive space 383 4 l inear series on curves with universa l s ingular i t ies 386 5 l inear series and project ive embedd ings of general curves 391 6 embedd ings of general cuspidal curves 400 7 degenera t ion of schuber t in tersect ions some special cases 403 8 degene ra t ion of schuber t in tersect ions ma in results 407 9 g c is reduced for general cuspidal r a t iona l c 414

on the projective normality of line bundles of extremal degree we propose a new method using deformation theory to study the maximal rank conjecture for line bundles of extremal degree which can be viewed as the first case to test the conjecture we prove that maximal rank conjecture holds by our new method

equipartitioning common domains of non atomic measures let h a1 an denote the harmonic mean n a1 1 a 1 1 of positive num bers a1 all and extend this definition by continuity to the non negative case by setting h a l an 0 if ai 0 for some i n recall that a measure j1 is non atomic if every measurable set a of strictly positive measure has a measur able subset b a satisfying j1 a j1 b o a measurable partition of a measure space x gf is a collection of disjoint subsets at an of x satisfying a egf n for all i and uai x throughout this paper measure means countably i 1 additive non negative measure and 11j1 11 is the total mass or total variation of the measure j1 the main purpose is to prove the following theorem

maximin share and minimax envy in fair division problems for fair division or cake cutting problems with value functions which are normalized positive measures i e the values are probability measures maximin share and minimax envy inequalities are derived for both continuous and discrete measures the tools used include classical and recent basic convexity results as well as ad hoc constructions examples are given to show that the envyminimizing criterion is not pareto optimal even if the values are mutually absolutely continuous in the discrete measure case sufficient conditions are obtained to guarantee the existence of envy free partitions

analysis of the projection operator method inconsistencies and their removal it is pointed out that the application of the projection operator method leads to the violation of a consistency condition if the so called born approximation is made in the nonmarkovian master equation since only the validity of this consistency condition guarantees obtaining non negative probabilities its violation makes the applicability of the whole method in this approximation meaningless therefore it is demonstrated that an additional markov approximation restores the validity of the consistency condition in the master equation and thus re establishes the applicability of the projection operator method furthermore the possibility of obtaining a self consistent projection operator method is also examined

families of nodal curves on projective threefolds and their regularity via postulation of nodes the main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds precisely given x a smooth projective threefold e a rank two vector bundle on x l a very ample line bundle on x and k 0 0 integers and denoted by v e l k the subscheme of p h 0 e l k parametrizing global sections of e l k whose zero loci are irreducible and nodal curves on x we present a new cohomological description of the tangent space t s v e l k at a point s 2 v e l k this description enable us to determine effective and uniform upper bounds for which are linear polynomials in k such that the family v e lk is smooth and of the expected dimension regular for short the almost sharpness of our bounds is shown by some interesting examples furthermore when x is assumed to be a fano or a calaby yau threefold we study in detail the regularity property of a point s 2 v e lk related to the postulation of the nodes of its zero locus c v s x roughly speaking when the nodes of c are assumed to be in general position either on x or on an irreducible divisor of x having at worst log terminal singularities or to lie on a l c i and subcanonical curve in x we find upper bounds onwhich are respectively cubic quadratic and linear polynomials in k ensuring the regularity of v e l k at s finally when x p 3 we also discuss some interesting geometric properties of the curves given by sections parametrized by v e ox k

a sharp partitioning inequality for non atomic probability measures based on the mass of the infimum of the measures if 1 are non atomic probability measures on the same measurable space s then there is an measurable partition ai i 1n of s so that i ai n 1 m 1 for all i 1 n where m left mathop lambda limits i 1 n mu i right is the total mass of the largest measure dominated by each of the i s moreover this bound is attained for all n 1 and all m in 0 1 this result is an analog of the bound n 1 m 1of elton et al 5 based on the mass m of the supremum of the measures each gives a quantative generalization of a well known cake cutting inequality of urbanik 10 and of dubins and spanier 2