the monodromy of weierstrass points we show that the monodromy of the family of curves riemann surfaces acts as the full symmetric group on the weierstrass points of a general curve the proof uses a degeneration to certain reducible curves and the theory of limit series developed in our 1986 1987a b some of the monodromy is actually constructed by fixing a reducible curve and varying its canonical series

geometry of curves with exceptional secant planes linear series along the general curve we study linear series on a general curve of genus g whose images are exceptional with regard to their secant planes working in the framework of an extension of brill noether theory to pairs of linear series we prove that a general curve has no linear series with exceptional secant planes in a very precise sense whenever the total number of series is finite next we partially solve the problem of computing the number of linear series with exceptional secant planes in a one parameter family in terms of tautological classes associated with the family by evaluating our hypothetical formula along judiciously chosen test families as an application we compute the number of linear series with exceptional secant planes on a general curve equipped with a one dimensional family of linear series we pay special attention to the extremal case of d secant d 2 planes to 2d 1 dimensional series which appears in the study of hilbert schemes of points on surfaces in that case our formula may be rewritten in terms of hypergeometric series which allows us both to prove that it is nonzero and to deduce its asymptotics in d

the birational type of the moduli space of even spin curves abstract we determine the kodaira dimension of the moduli space s g of even spin curves for all g precisely we show that s g is of general type for g 8 and has negative kodaira dimension for g 8

dimension counts for limit linear series on curves not of compact type we first prove a generalized brill noether theorem for linear series with prescribed multivanishing sequences on smooth curves we then apply this theorem to prove that spaces of limit linear series have the expected dimension for curves of pseudocompact type whenever the gluing conditions in the definition of limit linear series impose the maximal codimension finally we investigate these gluing conditions in specific families of curves showing expected dimension in several cases each with different behavior one of these families sheds new light on the work of cools draisma payne and robeva in tropical brill noether theory and suggests directions of further work in that setting

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

koszul modules and green s conjecture we prove a strong vanishing result for finite length koszul modules and use it to derive green s conjecture for every g cuspidal rational curve over an algebraically closed field mathbf k with text char mathbf k 0 or text char mathbf k ge frac g 2 2 as a consequence we deduce that the general canonical curve of genus g satisfies green s conjecture in this range our results are new in positive characteristic whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by voisin our strategy involves establishing two key results of independent interest 1 we describe an explicit characteristic independent version of hermite reciprocity for mathfrak s mathfrak l 2 representations 2 we completely characterize in arbitrary characteristics the non vanishing behavior of the syzygies of the tangential variety to a rational normal curve

the geometry of the moduli space of odd spin curves the spin moduli spacesg is the parameter space of theta characteristics spin structures on stable curves of genus g it has two connected components s g ands g depending on the parity of the spin structure we establish a complete birational classication by kodaira dimension of the odd components g of the spin moduli space we show thats g is uniruled for g 12 and even unirational for g 8 in this range introducing the concept of cluster for the mukai variety whose one dimensional linear sections are general canonical curves of genus g we construct new birational models ofs g these we then use to explicitly describe the birational structure of s g for instance s 8 is birational to a locally trivial p 7 bundle over the moduli space of elliptic curves with seven pairs of marked points for g 12 we prove thats g is a variety of general type in genus 12 this requires the construction of a counterexample to the slope conjecture on eective divisors on the moduli space of stable curves of genus 12

limit weierstrass points on nodal reducible curves in the 1980s d eisenbud and j harris posed the following question what are the limits of weierstrass points in families of curves degenerating to stable curves not of compact type in the present article we give a partial answer to this question we consider the case where the limit curve has components intersecting at points in general position and where the degeneration occurs along a general direction for this case we compute the limits of weierstrass points of any order however for the usual weierstrass points of order one we need to suppose that all of the components of the limit curve intersect each other

l2 index formulae for perturbed dirac operators the callias index theorem is generalized from the euclidean case to certain spin manifolds with warped ends making use of certain index preserving deformations

case control studies with contaminated controls abstract this paper considers inference about a parametric binary choice model when the data consist of two distinct samples the first is a random sample from the people who made choice 1 say with all relevant covariates completely observed the second is a random sample from the whole population with only the covariates observed this is called a contaminated sampling scheme an example might be where we have a random sample of female labor force participants and their covariate values and a second random sample of working age women with covariates whose participant status is unknown we consider the cases in which the fraction of the population making choice 1 is known and that in which it is not for both cases we give semiparametrically efficient procedures for estimating the choice model parameters

the bayesian bootstrap the bayesian bootstrap is the bayesian analogue of the bootstrap instead of simulating the sampling distribution of a statistic estimating a parameter the bayesian bootstrap simulates the posterior distribution of the parameter operationally and inferentially the methods are quite similar because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions neither method should be applied without some consideration of the reasonableness of these model assumptions in this sense neither method is a true bootstrap procedure yielding inferences unaided by external assumptions

linear series on a stable curve of compact type and relations among brill noether loci abstract let c be a stable curve of compact type such that two smooth curves with general moduli are bridged by a chain of n elliptic curves which will be called a tcbe curve in the present paper we investigate the existence of a smoothable limit linear series on c whose brill noether number satisfies n 0 through this work we find some relationships between a family of tcbe curves of genus g and a brill noether locus in the moduli space of stable curves of genus g further these results show some relations among brill noether loci

linear series on metrized complexes of algebraic curves a metrized complex of algebraic curves over an algebraically closed field kappa is roughly speaking a finite metric graph gamma together with a collection of marked complete nonsingular algebraic curves c v over kappa one for each vertex v of gamma the marked points on c v are in bijection with the edges of gamma incident to v we define linear equivalence of divisors and establish a riemann roch theorem for metrized complexes of curves which combines the classical riemann roch theorem over kappa with its graph theoretic and tropical analogues from amini and caporaso adv math 240 1 23 2013 baker and norine adv math 215 2 766 788 2007 gathmann and kerber math z 259 1 217 230 2008 and mikhalkin and zharkov tropical curves their jacobians and theta functions contemporary mathematics 203 231 2007 providing a common generalization of all of these results for a complete nonsingular curve x defined over a non archimedean field mathbb k together with a strongly semistable model mathfrak x for x over the valuation ring r of mathbb k we define a corresponding metrized complex mathfrak c mathfrak x of curves over the residue field kappa of mathbb k and a canonical specialization map tau mathfrak c mathfrak x from divisors on x to divisors on mathfrak c mathfrak x which preserves degrees and linear equivalence we then establish generalizations of the specialization lemma from baker algebra number theory 2 6 613 653 2008 and its weighted graph analogue from amini and caporaso adv math 240 1 23 2013 showing that the rank of a divisor cannot go down under specialization from x to mathfrak c mathfrak x as an application we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to eisenbud and harris invent math 85 337 371 1986 using this link we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove among other things that any degeneration of a mathfrak g r d in a regular family of semistable curves is a limit mathfrak g r d on the special fiber

callias index theorem elliptic boundary conditions and cutting and gluing it is shown that elliptic boundary conditions play the same role in callias index theorem as spectral boundary conditions do in the atiyah patodi singer index theorem this is used to generalize callias index theorem to arbitrary complete spin manifolds

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

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

an l2 index theorem for dirac operators on s1 r3 abstract an expression is found for the l2 index of a dirac operator coupled to a connection on a un vector bundle over s1 r 3 boundary conditions for the connection are given which ensure the coupled dirac operator fredholm callias index theorem is used to calculate the index when the connection is independent of the coordinate on s1 an excision theorem due to gromov lawson and anghel reduces the index theorem to this special case

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

limit canonical systems on curves with two components in the 80 s d eisenbud and j harris considered the following problem what are the limits of weierstrass points in families of curves degenerating to stable curves but for the case of stable curves of compact type treated by them this problem remained wide open since then in the present article we propose a concrete approach to this problem and give a quite explicit solution for stable curves with just two irreducible components meeting at points in general position

when ramification points meet this paper contains three applications of the technique of limit series our 1986 to the theory of ramification of linear series on smooth curves and curves of compact type overc

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

limit linear series and ranks of multiplication maps we develop a new technique for studying ranks of multiplication maps for linear series via limit linear series and degenerations to chains of genus 1 curves we use this approach to prove a purely elementary criterion for proving cases of the maximal rank conjecture and then apply the criterion to several ranges of cases giving a new proof of the case of quadrics and also treating several families in the case of cubics our proofs do not require restrictions on direction of approach so we recover new information on the locus in the moduli space of curves on which the maximal rank condition fails

bayes wesml posterior inference from choice based samples abstract in this paper i show that the weighted exogenous sampling likelihood wesml estimator for choice based samples due to manski and lerman 1977 can be given a bayesian interpretation under a multinomial model for the data with improper dirichlet priors bayesian posterior distributions of choice model parameters are computed to study the gains from choice based sampling and the effect of knowledge of population marginal choice probabilities