brauer groups and quotient stacks a natural question is to determine which algebraic stacks are quotient stacks in this paper we give some partial answers and relate it to the question of whether for a scheme x the natural map from the brauer group equivalence classes of azumaya algebras to the cohomological brauer group the torsion subgroup of h 2 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i m is surjective

on differential photometric reconstruction for unknown isotropic brdfs this paper presents a comprehensive theory of photometric surface reconstruction from image derivatives in the presence of a general unknown isotropic brdf we derive precise topological classes up to which the surface may be determined and specify exact priors for a full geometric reconstruction these results are the culmination of a series of fundamental observations first we exploit the linearity of chain rule differentiation to discover photometric invariants that relate image derivatives to the surface geometry regardless of the form of isotropic brdf for the problem of shape from shading we show that a reconstruction may be performed up to isocontours of constant magnitude of the gradient for the problem of photometric stereo we show that just two measurements of spatial and temporal image derivatives from unknown light directions on a circle suffice to recover surface information from the photometric invariant surprisingly the form of the invariant bears a striking resemblance to optical flow however it does not suffer from the aperture problem this photometric flow is shown to determine the surface up to isocontours of constant magnitude of the surface gradient as well as isocontours of constant depth further we prove that specification of the surface normal at a single point completely determines the surface depth from these isocontours in addition we propose practical algorithms that require additional initial or boundary information but recover depth from lower order derivatives our theoretical results are illustrated with several examples on synthetic and real data

log canonical models for the moduli space of curves the first divisorial contraction in this paper we initiate our investigation of log canonical models for m g as we decrease from 1 to 0 we prove that for the first critical value 9 11 the log canonical model is isomorphic to the moduli space of pseudostable curves which have nodes and cusps as singularities we also show that 7 10 is the next critical value i e the log canonical model stays the same in the interval 7 10 9 11 in the appendix we develop a theory of log canonical models of stacks that explains how these can be expressed in terms of the coarse moduli space

log canonical singularities are du bois a recurring difficulty in the minimal model program mmp is that while log terminal singularities are quite well behaved for instance they are rational log canonical singularities are much more complicated they need not even be cohenmacaulay the aim of this paper is to prove that as conjectured in kol92 1 13 log canonical singularities are du bois the concept of du bois singularities abbreviated as db was introduced by steenbrink in ste83 as a weakening of rationality it is not clear how to define du bois singularities in positive characteristic so we work over a field of characteristic 0 throughout the paper the precise definition is rather involved see 1 10 but our main applications rely only on the following consequence

towards a classification of modular compactifications of mathcal m _ g n the moduli space of smooth curves admits a beautiful compactification mathcal m g n subset overline mathcal m g n by the moduli space of stable curves in this paper we undertake a systematic classification of alternate modular compactifications of mathcal m g n let mathcal u g n be the non separated moduli stack of all n pointed reduced connected complete one dimensional schemes of arithmetic genus g when g 0 mathcal u 0 n is irreducible and we classify all open proper substacks of mathcal u 0 n when g 1 mathcal u g n may not be irreducible but there is a unique irreducible component mathcal v g n subset mathcal u g n containing mathcal m g n we classify open proper substacks of mathcal v g n satisfying a certain stability condition

a characterization of rational singularities is given bythe reynolds operator a well known and widely used theoremstates that in characteristic 0 canonicalsingularities are cohen macaulay and therefore rational see e and kmm the original proofs are based on a very clever use of grothendieck duality simulta neously for a resolution and its restriction onto the exceptional divisor and on a doubleloop induction kollar gave a simpler proof in k2 11 without using derived cate gories but still relying on a technically hard vanishing theorem recently kollar andmori found a simple proof allowing nonempty boundaries they do not use derivedcategories either but restrict to the projective case see km 5 18 these proofs areingenious but one would like to have a simple natural proof at least in the classical case when the boundary is empty as an application of theorem1 a simple proof is given here in the classical case but without the projective assumption this proof seems even simpler than that ofkollar and mori derived categories and grothendieck duality are used but in sucha simple way that one is tempted to say that this proof is the most natural one notealso that everything used here was already available when the question was raised forthe rst time a statement similar to theorem 1 was given in k2 3 12 some ideas of the

indoor scene reconstruction using near light photometric stereo we propose a novel framework for photometric stereo ps under low light conditions using uncalibrated near light illumination it operates on free form video sequences captured with a minimalistic and affordable setup we address issues such as albedo variations shadowing perspective projections and camera noise our method uses specular spheres detected with a perspective correcting hough transform to robustly triangulate light positions in the presence of outliers via a least squares approach furthermore we propose an iterative reweighting scheme in combination with an ell p norm minimizer to robustly solve the calibrated near light ps problem in contrast to other approaches our framework reconstructs depth albedo relative to light source intensity and normals simultaneously and is demonstrated on synthetic and real world scenes

on the direct summand conjecture and its derived variant andre recently gave a beautiful proof of hochster s direct summand conjecture in commutative algebra using perfectoid spaces his two main results are a generalization of the almost purity theorem the perfectoid abhyankar lemma and a construction of certain faithfully flat extensions of perfectoid algebras where discriminants acquire all p power roots in this paper we explain a quicker proof of hochster s conjecture that circumvents the perfectoid abhyankar lemma instead we prove and use a quantitative form of scholze s hebbarkeitssatz the riemann extension theorem for perfectoid spaces the same idea also leads to a proof of a derived variant of the direct summand conjecture put forth by de jong

framed sheaves on projective stacks abstract given a normal projective irreducible stack x over an algebraically closed field of characteristic zero we consider framed sheaves on x i e pairs e e where e is a coherent sheaf on x and e is a morphism from e to a fixed coherent sheaf f after introducing a suitable notion of semi stability we construct a projective scheme which is a moduli space for semistable framed sheaves with fixed hilbert polynomial and an open subset of it which is a fine moduli space for stable framed sheaves if x is a projective irreducible orbifold of dimension two and f a locally free sheaf on a smooth divisor d x satisfying certain conditions we consider d f framed sheaves i e framed sheaves e e with e a torsion free sheaf which is locally free in a neighbourhood of d and e d an isomorphism these pairs are stable for a suitable choice of a parameter entering the semi stability condition and of the polarization of x this implies the existence of a fine moduli space parameterizing isomorphism classes of d f framed sheaves on x with fixed hilbert polynomial which is a quasi projective scheme in an appendix we develop the example of stacky hirzebruch surfaces this is the first paper of a project aimed to provide an algebro geometric approach to the study of gauge theories on a wide class of 4 dimensional riemannian manifolds by means of framed sheaves on stacky compactifications of them in particular in a subsequent paper 20 these results are used to study gauge theories on ale spaces of type a k

pfaffian presentations of elliptic normal curves we investigate certain alternating matrices of linear forms whose pfaffians generate the homogeneous ideal of an elliptic normal curve or one of its higher secant varieties

bayesian helmholtz stereopsis with integrability prior helmholtz stereopsis is a 3d reconstruction method uniquely independent of surface reflectance yet its sub optimal maximum likelihood formulation with drift prone normal integration limits performance via three contributions this paper presents a complete novel pipeline for helmholtz stereopsis first we propose a bayesian formulation replacing the maximum likelihood problem by a maximum a posteriori one second a tailored prior enforcing consistency between depth and normal estimates via a novel metric related to optimal surface integrability is proposed third explicit surface integration is eliminated by taking advantage of the accuracy of prior and high resolution of the coarse to fine approach the pipeline is validated quantitatively and qualitatively against alternative formulations reaching sub millimetre accuracy and coping with complex geometry and reflectance

equations of 1 d polarized abelian surfaces in this paper we study the equations of projectively embedded abelian surfaces with a polarization of type 1 d classical results say that given an ample line bundle l on an abelian surface a the line bundle l n is very ample for n 3 and furthermore in case n is even and n 4 the generators of the homogeneous ideal ia of the embedding of a via l n are all quadratic a possible choice for a set of generators of ia are the riemann theta relations on the other hand much less is known about embeddings via line bundles l of type 1 d that is line bundles l which are not powers of another line bundle on a it is well known that if d 5 and a is a general abelian surface then l is very ample while l can never be very ample for d 5 however even if d 5 l may not be very ample for special abelian surfaces we will restrict our attention in what follows only to the general abelian surface and wish to know what form the equations take for such a projectively embedded abelian surface a few special cases are well documented in the literature d 4 in which case the general surface is a singular octic in p3 cf blvs and d 5 in which case the abelian surface is described as the zero set of a section of the horrocks mumford bundle hm whereas its homogeneous ideal is generated by 3 heisenberg invariant quintics and 15 sextics cf ma also recent work by manolache and schreyer ms and by ranestad ra provides a description of the equations and syzygies in the case d 7

the hilbert stack let x s be a morphism of algebraic stacks that is locally of finite presentation with affine stabilizers we prove that there is an algebraic s stack the hilbert stack parameterizing proper a

optimal piezoelectric control of a plate subject to time dependent boundary moments and forcing function for vibration damping active vibration control problem for a rectangular plate subject to moment boundary conditions and forcing function is solved by means of a maximum principle the control is exercised by a patch actuator and the solution is formulated using an adjoint variable leading to a coupled boundary initial terminal value problem the application of the maximum principle yields the optimal control expression as well as the explicit solution for a simply supported plate the objective functional to be minimized is defined as a quadratic functional of displacement and velocity and also includes a penalty in terms of control voltage applied to the piezoelectric patch actuator the penalty term limits the amount of control energy spent during the control process numerical results are presented to assess the effect of the optimal control algorithm

prehomogeneous vector spaces and field extensions letk be an infinite field of characteristic not equal to 2 3 5 in this paper we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of galois extensions ofk which are splitting fields of equations of certain degrees and prove that the inverse image of this map corresponds bijectively with conjugacy classes of galois homomorphisms

efficient photometric stereo using kernel regression photometric stereo estimates surface normals from multiple images captured under different light directions using a fixed camera to deal with non lambertian reflections the recent photometric stereo methods employ iterative or optimization frameworks that are computationally expensive this paper proposes an efficient photometric stereo method using kernel regression which can be transformed to an eigendecomposition problem the kernel parameter is variable for each surface point so that it can cope with the variety of general reflectances the best kernel parameter is automatically determined by leave one out cross validation to improve computational efficiency the leave one out process is accelerated by fast matrix computation and proper normal initialization the proposed photometric stereo method is extensively evaluated on synthetic and real surfaces with various reflectances experimental results validate that the method is computationally efficient and achieves the state of the art accuracy in surface normal estimation

on generic identifiability of symmetric tensors of subgeneric rank we prove that the general symmetric tensor in sdcn 1 of rank r is identifiable provided that r is smaller than the generic rank that is its waring decomposition as a sum of r powers of linear forms is unique only three exceptional cases arise all of which were known in the literature our original contribution regards the case of cubics d 3 while for d 4 we rely on known results on weak defectivity by ballico ciliberto chiantini and mella

application of active piezoelectric patches in controlling the dynamic response of a thin rectangular plate under a moving mass abstract the governing differential equation of motion for an undamped thin rectangular plate with a number of bonded piezoelectric patches on its surface and arbitrary boundary conditions is derived using hamilton s principle a moving mass traveling on an arbitrary trajectory acts as an external excitation for the system the effect of the moving mass inertia is considered using all the out of plane translational acceleration components the method of eigenfunction expansion is used to transform the equation of motion into a number of coupled ordinary differential equations a classical closed loop optimal control algorithm is employed to suppress the dynamic response of the system determining the required voltage of each piezoactuator at any time interval in a numerical example for a simply supported square plate under two different loading paths the effect of the mass velocity and mass weight of the moving load on the dynamic behavior of the uncontrolled system is investigated the results show that depending on the path of the moving mass the inertia effect is very important causing different behaviors of the system in addition the number of vibrational modes involved in determining the dynamic response of the system is crucial the inertia effect is more important for an orbiting mass loading case compared to the case in which the moving mass is traversing the plate on a straight line a number of equally spaced piezo patches are used on the lower surface of the plate to control the displacement of the center point of the plate the implemented control mechanism proves to be very efficient in suppressing the near resonant dynamic response of the system requiring fairly low levels of voltage for each patch increasing the area of the employed piezo patches would reduce the required maximum voltage for controlling the response of the system

moduli spaces of weighted pointed stable curves abstract a weighted pointed curve consists of a nodal curve and a sequence of marked smooth points each assigned a number between zero and one a subset of the marked points may coincide if the sum of the corresponding weights is no greater than one we construct moduli spaces for these objects using methods of the log minimal model program and describe the induced birational morphisms between moduli spaces as the weights are varied in the genus zero case we explain the connection to geometric invariant theory quotients of points in the projective line and to compactifications of moduli spaces studied by kapranov keel and losev manin

fukaya categories of the torus and dehn surgery this paper is a companion to the authors forthcoming work extending heegaard floer theory from closed 3 manifolds to compact 3 manifolds with two boundary components via quilted floer cohomology we describe the first interesting case of this theory the invariants of 3 manifolds bounding regarded as modules over the fukaya category of the punctured 2 torus we extract a short proof of exactness of the dehn surgery triangle in heegaard floer homology we show that a structures on the graded algebra a formed by the cohomology of two basic objects in the fukaya category of the punctured 2 torus are governed by just two parameters m6 m8 extracted from the hochschild cohomology of a for the fukaya category itself m6 0

the closed form solution for the forced vibration of non uniform plates with distributed time dependent boundary conditions abstract the closed form solution for the forced vibration of a non uniform plate with distributed time dependent boundary conditions is obtained three levy type solutions for a plate with different boundary conditions are studied the two dimensional system is transformed so that it becomes a one dimensional one by taking a general change of the dependent variable with shifting functions the one dimensional system is further transformed so that it becomes a system composed of a non homogeneous governing differential equation and four homogeneous boundary conditions the self adjointness and the orthogonality condition for the eigenfunctions of the further transformed system with elastic boundary conditions are derived consequently the method of separation of variables can be used to solve the transformed system the shifting functions expressed in terms of the four fundamental solutions of the transformed system instead of the fifth degree polynomials taken by mindlin goodman are derived the physical meanings of these shifting functions are explored its application to the vibration control of a non uniform plate with boundary inputs is investigated

classical solutions of forced vibration of rectangular plate driven by displacement boundary conditions forced vibration of a rectangular plate excited by displacement boundary conditions is investigated the system of boundary conditions is decomposed into two fundamental types of sub systems which involves only one side or one corner the necessary conditions for the solutions to be of classical sense are discussed the transformation is designed to convert the problems to those of forced vibration by body forces with homogeneous boundary conditions the closed form solutions are derived for these two types of problems using fourier series additional conditions are proposed to the boundary functions so that the differentiated series converges uniformly and consequently the solutions are verified to satisfy the differential equations in a classical sense with continuous derivatives

f singularities via alterations we give characterizations of test ideals and f rational singularities via regular alterations formally the descriptions are analogous to standard characterizations of multiplier ideals and rational singularities in characteristic zero via log resolutions lastly we establish nadel type vanishing theorems up to finite maps for test ideals and further demonstrate how these vanishing theorems may be used to extend sections

a modular compactification of m_ 1 n from a_ infty structures we show that a certain moduli space of minimal a structures coincides with the modular compactification m1 n n 1 of m1 n constructed by smyth in 22 in addition we describe these moduli spaces and the universal curves over them by explicit equations prove that they are normal and gorenstein show that their picard groups have no torsion and that they have rational singularities if and only if n 11

pretwisted nonuniform beams with time dependent elastic boundary conditions the governing differential equations with the nonhomogeneous time dependent elastic boundary conditions for the coupled bending bending vibration of a pretwisted nonuniform beam are derived by hamilton s principle by taking a general change of dependent variable with shifting functions the original system is transformed to be a system composed of two nonhomogeneous governing differential equations and eight homogeneous boundary conditions consequently the method of separation ofvariables can beused to solvethetransformed problem the physical meanings of these shifting functions are explored the orthogonality condition for the eigenfunctions of a pretwistednonuniformbeamwithelasticboundaryconditionsisalsoderived thestiffnessmatrixforanonuniform beam with arbitrary pretwist is derived as the coefe cients of the matrix can be integrated analytically the exact stiffness matrix is therefore obtained the relation between the shifting functions and the stiffness matrix is derived the ine uence of the pretwist angle on the dynamic response of the beam is studied the vibration control of a pretwist beam with boundary inputs is investigated

moduli of curves grobner bases and the krichever map abstract we study moduli spaces of possibly non nodal curves c p 1 p n of arithmetic genus g with n smooth marked points equipped with nonzero tangent vectors such that o c p 1 p n is ample and h 1 o c a 1 p 1 a n p n 0 for given integer weights a a 1 a n such that a i 0 and a i g we show that each such moduli space u g n n s a is an affine scheme of finite type and the krichever map identifies it with the quotient of an explicit locally closed subscheme of the sato grassmannian by the free action of the group of changes of formal parameters we study the git quotients of u g n n s a by the natural torus action and show that some of the corresponding stack quotients give modular compactifications of m g n with projective coarse moduli spaces more generally using similar techniques we construct moduli spaces of curves with chains of divisors supported at marked points with prescribed number of sections which in the case n 1 corresponds to specifying the weierstrass gap sequence at the marked point

moduli spaces of nonspecial pointed curves of arithmetic genus 1 in this paper we study the moduli stack mathcal u 1 n ns of curves of arithmetic genus 1 with n marked points forming a nonspecial divisor in polishchuk a modular compactification of mathcal m 1 n from a infty structures arxiv 1408 0611 this stack was realized as the quotient of an explicit scheme widetilde mathcal u 1 n ns affine of finite type over mathbb p n 1 by the action of mathbb g m n our main result is an explicit description of the corresponding git semistable loci in widetilde mathcal u 1 n ns this allows us to identify some of the git quotients with some of the modular compactifications of mathcal m 1 n defined in smyth invent math 192 459 503 2013 compos math 147 3 877 913 2011

pluricanonical maps of stable log surfaces abstract stable surfaces and their log analogues are the type of varieties naturally occurring as boundary points in moduli spaces we extend classical results of kodaira and bombieri to this more general setting if x is a stable log surface with reduced boundary possibly empty and i is its global index then 4 i k x is base point free and 8 i k x is very ample these bounds can be improved under further assumptions on the singularities or invariants for example 5 k x is very ample if x has semi canonical singularities