estimates for measures of lower dimensional sections of convex bodies abstract we present an alternative approach to some results of koldobsky on measures of sections of symmetric convex bodies which allows us to extend them to the not necessarily symmetric setting we prove that if k is a convex body in r n with 0 int k and is a measure on r n with a locally integrable non negative density g on r n then k c n k k max f g n n k k f k k n for every 1 k n 1 also if is even and log concave and if k is a symmetric convex body in r n and d is a compact subset of r n such that k f d f for all f g n n k then k c k l n k k d where l s is the maximal isotropic constant of a convex body in r s our method employs a generalized blaschke petkantschin formula and estimates for the dual affine quermassintegrals

on the local equatorial characterization of zonoids and intersection bodies abstract in this paper we show that there is no local equatorial characterization of zonoids in odd dimensions this gives a negative answer to the conjecture posed by w weil in 1977 and shows that the local equatorial characterization of zonoids may be given only in even dimensions in addition we prove a similar result for intersection bodies and show that there is no local characterization of these bodies

inequalities between projection functions of convex bodies we obtain injectivity results for certain integral transforms on grassmann manifolds these are used to solve the intermediate cases of a problem of shephard and to give other applications to projection functions of convex bodies of revolution

busemann s intersection inequality in hyperbolic and spherical spaces abstract busemann s intersection inequality asserts that the only maximizers of the integral s n 1 k n d among all convex bodies of a fixed volume in r n are centered ellipsoids we study this question in the hyperbolic and spherical spaces as well as general measure spaces

generalized intersection bodies abstract we study the structures of two types of generalizations of intersection bodies and the problem of whether they are in fact equivalent intersection bodies were introduced by lutwak and played a key role in the solution of the busemann petty problem a natural geometric generalization of this problem considered by zhang led him to introduce one type of generalized intersection bodies a second type was introduced by koldobsky who studied a different analytic generalization of this problem koldobsky also studied the connection between these two types of bodies and noted that an equivalence between these two notions would completely settle the unresolved cases in the generalized busemann petty problem we show that these classes share many identical structural properties proving the same results using integral geometry techniques for zhang s class and fourier transform techniques for koldobsky s class using a functional analytic approach we give several surprising equivalent formulations for the equivalence problem which reveal a deep connection to several fundamental problems in the integral geometry of the grassmann manifold

sections of convex bodies the generalized busemann petty problem asks if the volume of i dimensional central section of a centrally symmetric convex body in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is smaller than that of another such body is the volume of the body also smaller it is proved that the answer is negative if 2 i n the case of a 2 dimensional section remains open the proof uses techniques in functional analysis and radon transforms on grassmannians it also requires the notion of an i intersection body which generalizes the notion of an intersection body inequalities among the volumes of projection bodies polar projection bodies and their central sections are proved they are related to the maximal slice problem

components of moduli spaces of spin curves with the expected codimension we prove a conjecture of gavril farkas claiming that for all integers r ge 2 and g ge left begin array c r 2 2 end array right there exists a component of the locus mathcal s r g of spin curves with a theta characteristic l such that h 0 l ge r 1 and h 0 l equiv r 1 text mod 2 which has codimension left begin array c r 1 2 end array right inside the moduli space mathcal s g of spin curves of genus g

lp dual curvature measures abstract a new family of geometric borel measures on the unit sphere is introduced special cases include the l p surface area measures which extend the classical surface area measure of aleksandrov and fenchel jessen and l p integral curvature which extends alkesandrov s integral curvature in the l p brunn minkowski theory it also includes the dual curvature measures which are the duals of federer s curvature measures in the dual brunn minkowski theory this partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes

sections of convex bodies with symmetries abstract in this paper we study how certain symmetries of convex bodies affect their geometric properties in particular we consider the impact of symmetries generated by the block diagonal subgroup of orthogonal transformations generalizing complex and quaternionic convex bodies we conduct a systematic study of sections of bodies with symmetries of this type with the emphasis on problems of the busemann petty type and hyperplane inequalities the main role belongs to the class of intersection bodies with symmetries

the average shadowing property and strong ergodicity abstract let x be a compact metric space and f x x be a continuous map in this paper we prove that if f has the average shadowing property and the minimal points of f are dense in x then f is weakly mixing and totally strongly ergodic as applications we obtain that if f is a distal or lyapunov stable map having the average shadowing property then x is consisting of one point moreover we illustrate that the full shift has the average shadowing property

the cramer rao inequality for star bodies associated with each body k in euclidean n space mathbb r sp n is an ellipsoid gamma sb 2k called the legendre ellipsoid of k it can be defined as the unique ellipsoid centered at the body s center of mass such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of the body in an earlier paper the authors showed that corresponding to each convex body k subset mathbb r sp n is a new ellipsoid gamma sb 2 k that is in some sense dual to the legendre ellipsoid the legendre ellipsoid is an object of the dual brunn minkowski theory while the new ellipsoid gamma sb 2 k is the corresponding object of the brunn minkowski theory the present paper has two aims the first is to show that the domain of gamma sb 2 can be extended to star shaped sets the second is to prove that the following relationship exists between the two ellipsoids if k is a star shaped set then gamma sb 2 k subset gamma sb 2k with equality if and only if k is an ellipsoid centered at the origin this inclusion is the geometric analogue of one of the basic inequalities of information theory the cramer rao inequality

intersection bodies in r4 intersection bodies were introduced in 1988 by lutwak who found a close connection between those bodies and the well known 1956 busemann petty problem which asks whether origin symmetric convex bodies with larger central hyperplane sections also have greater volume the author has recently shown that an origin symmetric star bodykis an intersection body if and only if the function x 1kis a positive definite distribution where x k min a 0 x ak we use this result to prove that the unit balls of the spaces l4q 2 q are intersection bodies forn 5 the unit balls of the spaces lnq 2 q are not intersection bodies the technique of this paper allows to find precise expressions for the generating measures signed measures of these bodies the result that the unit cube in r4is an intersection body shows that the negative solution to the busemann petty problem forn 4 in 20 was incorrect

dual orlicz brunn minkowski theory dual orlicz lϕ affine and geominimal surface areas abstract this paper aims to develop the basic theory for the dual orlicz l affine and geominimal surface areas of star bodies basic properties of these new affine invariants are established moreover related orlicz affine isoperimetric cyclic santalo type and alexandrov fenchel type inequalities are obtained

firey shephard problems for homogeneous measures abstract we introduce a concept of firey projection function for measure which is an extension of l p projection body we consider the firey shephard problem for measures and connect it with l p projection body using this connection we give a negative answer to the firey shephard problem for homogeneous measures for all p 1 p n and n 2 moreover we also prove a measure comparison theorem

valuations and busemann petty type problems projection and intersection bodies define continuous and gl n contravariant valuations they played a critical role in the solution of the shephard problem for projections of convex bodies and its dual version for sections the busemann petty problem we consider the question whether fkfl implies v k v l where f is a homogeneous continuous operator on convex or star bodies which is an so n equivariant valuation important previous results for projection and intersection bodies are extended to a large class of valuations

grothendieck s pairing on component groups of jacobians let r be a discrete valuation ring with field of fractions k let ak be an abelian variety over k with dual ak denote by a and a the corresponding neron models and by a and a their component groups in gr exp vii ix grothendieck used the notion of biextension invented by mumford to investigate how the duality between ak and ak is reflected on the level of neron models in fact the essence of the relationship between a and a is captured by a bilinear pairing a a q z

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

p q mixed geominimal surface area and p q mixed affine surface area abstract in this paper the p q mixed geominimal surface area and the p q mixed affine surface area are introduced they are extensions of the l p geominimal surface area and the l p affine surface area respectively affine invariance under unimodular affine transformations and continuity for the p q mixed geominimal surface area and the p q mixed affine surface area are discussed the existence the uniqueness and the uniform boundness of the p q mixed petty bodies are obtained affine isoperimetric inequalities for the p q mixed geominimal surface area and the p q mixed affine surface area are established

dual orlicz brunn minkowski theory abstract in this paper a dual orlicz brunn minkowski theory is presented an orlicz radial sum and dual orlicz mixed volumes are introduced the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality are established the variational formula for the volume with respect to the orlicz radial sum is proved the equivalence between the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality is demonstrated orlicz intersection bodies are defined and the orlicz busemann petty problem is posed

on dual quermassintegrals of mixed intersection bodies in this paper it is shown that a family of inequalities involving mixed intersection bodies holds the busemann intersection inequality is the first of this family all of the members of this family are strengthened versions of classical inequalities between pairs of dual quermassintegrals of a star body

generalized intersection bodies are not equivalent abstract in a koldobsky a functional analytic approach to intersection bodies geom funct anal 10 2000 1507 1526 a koldobsky asked whether two types of generalizations of the notion of an intersection body are in fact equivalent the structures of these two types of generalized intersection bodies have been studied by the author in e milman generalized intersection bodies j funct anal 240 2 2006 530 567 providing substantial evidence for a positive answer to this question the purpose of this note is to construct a counter example which provides a surprising negative answer to this question in a strong sense this implies the existence of non trivial non negative functions in the range of the spherical radon transform and the existence of non trivial spaces which embed in l p for certain negative values of p

intersection bodies and valuations all gl n covariant star body valued valuations on convex polytopes are completely classified it is shown that there is a unique nontrivial such valuation this valuation turns out to be the so called intersection operator an operator that played a critical role in the solution of the busemann petty problem

the busemann petty problem in hyperbolic and spherical spaces abstract the busemann petty problem asks whether origin symmetric convex bodies in r n with smaller central hyperplane sections necessarily have smaller n dimensional volume it is known that the answer to this problem is affirmative if n 4 and negative if n 5 we study this problem in hyperbolic and spherical spaces

the average shadowing property and topological ergodicity in this note we prove that a lyapunov stable map having the average shadowing property from a compact metric space onto itself is topologically ergodic but it is not topologically weakly mixing

plucker forms and the theta map let rm su x r 0 be the moduli space of semistable vector bundles of rank r and trivial determinant over a smooth irreducible complex projective curve x the theta map theta r rm su x r 0 rightarrow bbb p n is the rational map defined by the ample generator of rm pic rm su x r 0 the main result of the paper is that theta r is generically injective if g gg r and x is general this partially answers the following conjecture proposed by beauville theta r is generically injective if x is not hyperelliptic the proof relies on the study of the injectivity of the determinant map d e wedge r h 0 e rightarrow h 0 det e for a vector bundle e on x and on the reconstruction of the grassmannian g r rm from a natural multilinear form associated to it defined in the paper as the pl ucker form the method applies to other moduli spaces of vector bundles on a projective variety x

a new ellipsoid associated with convex bodies it is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the legendre ellipsoid of classical mechanics sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid these inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid among other things this leads to a new approach to establishing ball s maximal shadows conjecture for symmetric convex bodies corresponding to each origin symmetric convex or more general subset of euclidean n space r there is a unique ellipsoid with the following property the moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1 dimensional subspace of r this ellipsoid is called the legendre ellipsoid of the convex set the legendre ellipsoid and its polar the binet ellipsoid are well known concepts from classical mechanics see milman and pajor mpa1 mpa2 lindenstrauss and milman lim and leichtweis le for some historical references it has slowly come to be recognized that along side the brunn minkowski theory there is a dual theory the nature of the duality between this dual brunn minkowski theory and the brunn minkowski theory is subtle and not yet understood it is easily seen that the legendre and binet ellipsoid is an object of this dual brunnminkowski theory this observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical legendre ellipsoid in the brunn minkowski theory it is the aim of this paper to demonstrate the existence of precisely this dual object in retrospect one may well wonder why research supported in part by nsf grant dms 9803261 typeset by ams tex 1 2 a new ellipsoid associated with convex bodies the new ellipsoid presented in this note was not discovered long ago the simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of l2 curvature in hand however the brunn minkowski theory was only recently extended to incorporate the new notion of lp curvature see l2 l3 a positive definite n n real symmetric matrix a generates an ellipsoid a in r defined by a x r x ax 1 where x ax denotes the standard inner product of x and ax in r associated with a star shaped about the origin set k r is its legedre ellipsoid 2k which is generated by the matrix mij k where mij k n 2 v k k ei x ej x dx with e1 en denoting the standard basis for r and v k denoting the ndimensional volume of k we will associate a new ellipsoid 2k with each convex body k r one approach to defining 2k without introducing new notation is to first define it for polytopes and then use approximation with respect to the hausdorff metric to extend the definition to all convex bodies suppose p r is a polytope that contains the origin in its interior let u1 un denote the outer unit normals to the faces of p let a1 an denote the areas i e n 1 dimensional volumes of the corresponding faces and let h1 hn denote the distances from the origin to the corresponding faces the ellipsoid 2p is generated by the matrix mij p where mij p 1 v p n l 1 al hl ei ul ej ul an alternate definition of the operator 2 will be given after additional notation is introduced the easily established affine nature of the operator 2 is formally stated in lemma 1 if k r is star shaped about the origin then for each gl n

volume of mixed bodies by using inequalities obtained for the volume of mixed bodies and the petty projection inequality sharp isoperimetric inequalities are derived for the projection measures quermassintegrale of a convex body these projection measure inequalities which involve mixed projection bodies zonoids are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body the inequality obtained for the volume of mixed bodies is also used to derive a form of the brunn minkowski inequality involving mixed bodies as an application inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies o introduction in 24 it was shown that the petty projection inequality 26 will yield extensions of the known isoperimetric inequalities between the volume of a convex body and its projection measures quermassintegrale in this article we show that the petty projection inequality can be used to obtain sharp isoperimetric inequalities corollary 5 20 which are strengthened forms of the classical inequalities between the projection measures of a convex body this is accomplished by using the so called mixed convex bodies introduced by firey 17 geometric inequalities are derived for the volume of mixed bodies theorems 4 1 and 4 3 which when combined with the petty projection inequality yield inequalities involving projection bodies zonoids which are related to the inequalities between the projection measures of a convex body the inequalities for mixed bodies are also used to obtain a version of the brunn minkowski inequality theorem 4 2 involving mixed bodies also derived are inequalities between the volume of convex bodies and their associated mixed projection integral theorems 6 4 and 6 4 that are extensions and improvements of some previously established inequalities since some of our results concern projection bodies a topic of interest to workers outside the area of geometric convexity we have restated a few basic definitions and standard results from the field of geometric inequalities in 1 we collect for quick reference some well known results concerning mixed volumes and mixed area measures which will be required later references for material presented in this section are bonnesen and fenchel 7 for mixed volumes busemann 9 and leichtweiss 20 the original works of aleksandrov 1 4 and fenchel and jessen 15 supplemented by the survey of schneider 28 are recommended references for mixed area measures 2 contains introductory material regarding blaschke addition received by the editors january 8 1985 1980 mcithematics subject classihecition primary 52a40 secondary 53a15 46b20

stability in the busemann petty and shephard problems abstract a comparison problem for volumes of convex bodies asks whether inequalities f k f l for all s n 1 imply that vol n k vol n l where k l are convex bodies in r n and f k is a certain geometric characteristic of k by linear stability in comparison problems we mean that there exists a constant c such that for every e 0 the inequalities f k f l e for all s n 1 imply that vol n k n 1 n vol n l n 1 n c e we prove such results in the settings of the busemann petty and shephard problems and their generalizations we consider the section function f k s k vol n 1 k and the projection function f k p k vol n 1 k where is the central hyperplane perpendicular to and k is the orthogonal projection of k to in these two cases we prove linear stability under additional conditions that k is an intersection body or l is a projection body respectively then we consider other functions f k which allow to remove the additional conditions on the bodies in higher dimensions

the complex busemann petty problem on sections of convex bodies abstract the complex busemann petty problem asks whether origin symmetric convex bodies in c n with smaller central hyperplane sections necessarily have smaller volume we prove that the answer is affirmative if n 3 and negative if n 4

a positive answer to the busemann petty problem in three dimensions we prove that in e3 the busemann petty problem concerning cen tral sections of centrally symmetric convex bodies has a positive answer together with other results this settles the problem in each dimension

minimal resolutions chow forms and ulrich bundles on k3 surfaces the minimal resolution conjecture mrc for points on a projective varietyx p predicts that the minimal graded free resolution of a general set x of points is as simple as the geometry of x allows originally the most studied case has been that when x p see epsw the general form of the mrc for subvarieties x p was formulated in mus and fmp the betti diagram of a large enough set x consisting of general points is obtained from the betti diagram of x by adding two rows indexed by u 1 and u where u is an integer depending on all differences bi 1 u 1 bi u are known and depend on the hilbert polynomial px and i u and see fmp the minimal resolution conjecture for general points on x predicts that

extremum problems for the cone volume functional of convex polytopes abstract lutwak yang and zhang defined the cone volume functional u over convex polytopes in r n containing the origin in their interiors and conjectured that the greatest lower bound on the ratio of this centro affine invariant u to volume v is attained by parallelotopes in this paper we give affirmative answers to the conjecture in r 2 and r 3 some new sharp inequalities characterizing parallelotopes in r n are established moreover a simplified proof for the conjecture restricted to the class of origin symmetric convex polytopes in r n is provided

star valuations and dual mixed volumes since its creation by brunn and minkowski what has become known as the brunn minkowski theory has provided powerful machinery to solve a broad variety of inverse problems with stereological data the machinery of the brunn minkowski theory includes mixed volumes of minkowski symmetrization techniques such as those of steiner and blaschke isoperimetric inequalities such as the brunn minkowski minkowski and aleksandrov fenchel inequalities integral transforms such as the cosine transform and important auxiliary bodies associated with these transforms such as minkowski s projection bodies schneider s recent book 22 on the brunn minkowski theory is the best available introduction to the subject while the brunn minkowski theory has proven to be of enormous value in answering inverse questions regarding projections of convex bodies onto subspaces the theory has been of little value in answering inverse questions with data regarding intersections with subspaces however recent advancements have been made in the development of a dual brunn minkowski theory 3 4 5 6 8 9 11 13 16 23 26 27 28 29 which has been tailored specifically for dealing with such questions in contrast to the brunn minkowski theory in the dual theory convex bodies are replaced by star shaped sets and projections onto subspaces are replaced by intersections with subspaces the machinery of the dual theory includes dual mixed volumes introduced by lutwak 15 16 dual isoperimetric inequalities 16 and important auxiliary bodies known as intersection bodies first introduced by busemann for special centered convex bodies in 1 later defined for star shaped sets by lutwak 16 see also 9 a comprehensive introduction to geometric tomography including the dual brunn minkowski theory may be found in gardner s book 17 unfortunately there still remain fundamental and foundational problems with the dual brunn minkowski theory one of the most beautiful and article no 0048

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

on the existence of components of the hilbert scheme with the expected number of moduli in the study of the geometry of families of algebraic curves one common tool is the comparison between the extrinsic geometry represented by properties of their projective embeddings and of the hilbert scheme and the intrinsic geometry represented by abstract properties and the moduli space this comparison is often best represented by the natural map hn g r mg from the hilbert scheme hn g r of smooth curves in ip r of degree n and genus g or the hurwitz scheme for r 1 the severi variety for r 2 to the moduli space of smooth curves of genus g the behaviour of is very strongly influenced by the nature of its fibers the brillnoether varieties now whilemg is an irreducible variety the hilbert scheme hn g r is in general reducible and has in fact many components for r 3 this difference has inspired many authors in the search of what could be a good component of the hilbert scheme one possible answer to such a question is to apply the above principle of comparison ser

irreducibility and components rigid in moduli of the hilbert scheme of smooth curves denote by mathcal h d g r the hilbert scheme of smooth curves that is the union of components whose general point corresponds to a smooth irreducible and non degenerate curve of degree d and genus g in mathbb p r a component of mathcal h d g r is rigid in moduli if its image under the natural map pi mathcal h d g r dashrightarrow mathcal m g is a one point set in this note we provide a proof of the fact that mathcal h d g r has no components rigid in moduli for g 0 and r 3 from which it follows that the only smooth projective curves embedded in mathbb p 3 whose only deformations are given by projective transformations are the twisted cubic curves in case r ge 4 we also prove the non existence of a component of mathcal h d g r rigid in moduli in a certain restricted range of d g 0 and r in the course of the proofs we establish the irreducibility of mathcal h d g 3 beyond the range which has been known before

on shadowing ordinary and ergodic the aim of this paper is to introduce the notion of ergodic shadowing for a continuous onto map which is equivalent to the map being topologically mixing and has the ordinary shadowing property in particular we deduce the chaotic behavior of a map with ergodic shadowing property moreover we define some kind of specification property and investigate its relation to the ergodic shadowing property

lutwak petty projection inequalities for minkowski valuations and their duals abstract lutwak s volume inequalities for polar projection bodies of all orders are generalized to polarizations of minkowski valuations generated by even zonal measures on the euclidean unit sphere this is based on analogues of mixed projection bodies for such minkowski valuations and a generalization of the notion of centroid bodies a new integral representation is used to single out lutwak s inequalities as the strongest among these families of inequalities which in turn are related to a conjecture on affine quermassintegrals in the dual setting a generalization of volume inequalities for intersection bodies of all orders by leng and lu is proved these results are related to grinberg s inequalities for dual affine quermassintegrals

gaussian measure of sections of convex bodies abstract in this paper we study properties of sections of convex bodies with respect to the gaussian measure we develop a formula connecting the minkowski functional of a convex symmetric body k with the gaussian measure of its sections using this formula we solve an analog of the busemann petty problem for gaussian measures

an analytic solution to the busemann petty problem on sections of convex bodies we derive a formula connecting the derivatives of parallel section functions of an origin symmetric star body in r n with the fourier transform of powers of the radial function of the body a parallel section function or ni 1 dimensional x ray gives the ni 1 dimensional volumes of all hyperplane sections of the body orthogonal to a given direction this formula provides a new characterization of intersection bodies in r n and leads to a unifled analytic solution to the busemann petty problem suppose that k and l are two origin symmetric convex bodies inr n such that the ni 1 dimensional volume of each central hyperplane section of k is smaller than the volume of the corresponding section of l is the n dimensional volume of k smaller than the volume of l in conjunction with earlier established connections between the busemann petty problem intersection bodies and positive deflnite distributions our formula shows that the answer to the problem depends on the behavior of the ni 2 nd derivative of the parallel section functions the a rmative answer to the busemann petty problem for n 4 and the negative answer for n 5 now follow from the fact that convexity controls the second derivatives but does not control the derivatives of higher orders

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

brill noether loci and the gonality stratification of m g for an irreducible smooth projective complex curve c of genus g the gonality defined as gon c min d z 1 there exists a gd on c is perhaps the second most natural invariant it gives an indication of how far c is from being rational in a way different from what the genus does for g 3 we consider the stratification of the moduli space mg of smooth curves of genus g given by gonality mg 2 m 1 g 3 m 1 g k mg

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

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

generalizations of the busemann petty problem for sections of convex bodies abstract we present generalizations of the busemann petty problem for dual volumes of intermediate central sections of symmetric convex bodies it is proved that the answer is negative when the dimension of the sections is greater than or equal to 4 for two three dimensional sections both negative and positive answers are given depending on the orders of dual volumes involved and certain cases remain open for bodies of revolution a complete solution is obtained in all dimensions

intersection bodies positive definite distributions and the busemann petty problem the 1956 busemann petty problem asks whether symmetric convex bodies with larger central hyperplane sections also have greater volume in 1988 lutwak introduced the concept of an intersection body which is closely related to the busemann petty problem we prove that an origin symmetric star body k in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is an intersection body if and only if x 1 k is a positive definite distribution on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i where x k min a 0 x ak we use this result to show that for every dimension n there exist polytopes in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i which are intersection bodies for example the cross polytope the unit ball of every subspace of l p 0 p 2 is an intersection body the unit ball of the space l n q 2 q n 5 using lutwak s connection with the busemann petty problem we present new counterexamples to the problem for n 5 and confirm the conjecture of meyer that the answer to the problem is affirmative if the smaller body is a polar projection body

mixed affine surface area various means over the unit sphere of powers of the curvature function of a convex body arise in geometric convexity for example both the surface area and the affine surface area of sufficiently smooth bodies can be expressed as such means the survey of gruber 111 on approximation of convex bodies contains a number of examples of means of powers of the curvature function arising in approximation problems see pp 137 141 143 and 1466148 of 11 i these means arise surprisingly in questions related to approximating convex bodies by polytopes see e g mcclure and vitale 18 schneider 24 and schneider and wieacker 27 an interesting conjecture of firey 9 p s also 26 p 2571 which was recently proved by gage lo for plane convex bodies is that a certain weighted mean of the curvature function lies between the arithmetic and harmonic means of the curvature function it turns out that there is also a close connection between various means of the curvature function of a convex body and some mixed volumes involving the body in this note we investigate means of the curvature function of a convex body which arise as coefficients in the expansion of the afftne surface area of outer and inner blaschke parallel bodies for a variety of reasons we will take a general approach to the study of these means an approach similar to that taken in 14 see also 4 pp 164 1671 in presenting dual mixed volumes let x denote the set of convex bodies compact convex sets with interior points in euclidean n space r we use h k to denote the support function of a convex body k i e

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

the busemann petty problem for arbitrary measures the busemann petty problem asks whether symmetric convex bodies in n with smaller n 1 dimensional volume of central hyperplane sections necessarily have smaller n dimensional volume the answer to this problem is affirmative for n 4 and negative for n 5 in this paper we generalize the busemann petty problem to essentially arbitrary measure in place of the volume we also present applications of the latter result by proving several inequalities concerning the measure of sections of convex symmetric bodies in n

lp affine isoperimetric inequalities the lp analogues of the petty projection inequality and the busemannpetty centroid inequality are established an affine isoperimetric inequality compares two functionals associated with convex or more general bodies where the ratio of the functionals is invariant under non degenerate linear transformations these isoperimetric inequalities are more powerful than their better known euclidean relatives

an extension of minkowski s theorem and its applications to questions about projections for measures abstract minkowski s theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body and the surface area measure determines a convex body uniquely up to a shift in this manuscript we prove an extension of minkowski s theorem consider a measure on r n with positive degree of concavity and positive degree of homogeneity we show that a surface area measure of a convex set k weighted with respect to determines a convex body uniquely up to measure zero we also establish an existence result under natural conditions including symmetry we apply this result to extend the solution to classical shephard s problem to do this we introduce a new notion which relates projections of convex bodies to a given measure and is a generalization of the lebesgue volume of a projection

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