Geometry, Algebra, Singularities, Combinatorics (GASC)
Monday, 12:15 PM, 511 LA
Organizers: AnaMaria Castravet, Tony Iarrobino, Ivan Martino, Egon Schulte, Alex Suciu and Chris Beasley (on leave)
Upcoming Events
Previous Events

Feb 13 2017 GASC Seminar "Canonical cycle classes on K3 surfaces and hyperKähler varieties" by Xiaolei Zhao
Canonical cycle classes on K3 surfaces and hyperKähler varieties
The Chow groups of algebraic cycles on algebraic varieties have many mysterious properties. For K3 surfaces, on the one...
Location: Lake Hall 
BernsteinSato polynomials of ideals of matrices
A decade ago, N. Budur, M. Mustata and M. Saito introduced "bfunctions of general varieties" (as opposed to the classical case of...
Location: Lake Hall 
Jan 30 2017 GASC Seminar "Extremal divisors from the strata of abelian differentials" by Scott Mullane
Extremal divisors from the strata of abelian differentials
An abelian differential defines a flat metric with singularities at its zeros and poles, such that the underlying Riemann surface...
Location: Lake Hall 
Topology of the tropical moduli spaces of curves
The moduli space of nmarked, genus g tropical curves is a cell complex that was identified in work of Abramovich  Caporaso  Payne with...
Location: Lake Hall 
Splines and a conjecture of Strang
Splines are an essential tool in analysis and applied math to deal with approximations and interpolation. We describe a general algebraic/combinatorial...
Location: Lake Hall 
Nov 28 2016 GASC Seminar "Measures of irrationality for hypersurfaces of large degree" by Brooke Ullery
Measures of irrationality for hypersurfaces of large degree
The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. There are a few...
Location: Lake Hall 
Nov 21 2016 GASC Seminar "Levi subgroup actions on Schubert varieties in the Grassmannian" by Reuven Hodges
Levi subgroup actions on Schubert varieties in the Grassmannian
Let L be the Levi part of the stabilizer Q in GL_N(C) (for left multiplication) of a Schubert variety X(w) in the...
Location: Lake Hall 
Nov 14 2016 GASC Seminar "Exotic deformation behavior: auxetic and expansive trajectories" by Ileana Streinu
Exotic deformation behavior: auxetic and expansive trajectories
In materials science, auxetic behavior refers to the rather counterintuitive property of a material becoming laterally wider...
Location: Lake Hall 
CohenMacaulay representation theory and singularities
For a finite group G, classical representation theory seeks to understand left modules over the group ring kG, where k is a field. One...
Location: Lake Hall 
Bases for cluster algebras
Cluster algebras were originally introduced to axiomatize part of the 'dual canonical basis' for the ring of functions on a Lie group, and similar patterns have...
Location: Lake Hall 
Primitive deformations of quantum pgroups
The classification of finitedimensional Hopf algebras in the field of characteristic zero has been investigated for decades with many fruitful...
Location: LAKE HALL 
Higgs Moduli on Hyperelliptic curves
Let C be a smooth projective variety (or, what is the same, compact Riemann surface) of genus g. One understands the behaviour of meromorphic functions...
Location: Lake Hall 
Oct 3 2016 GASC Seminar talk by Dhruv Ranganthan
Tropical curve counting in obstructed geometries
Abstract: Tropical geometry provides a number of combinatorial techniques to tackle questions in the enumerative geometry of toric...
Location: Lake Hall 
Sep 26 2016 GASC Seminar
"Topology of a center transversal theorem for the superRado depth", by Pavle V. M. Blagojevic (Free University, Berlin)
Abstract. A celebrated result of Dolnikov, and of Zivaljevic and...
Location: 511 Lake Hall 
Sep 19 2016 GASC Seminar Talk
"Complements of homotopy sphere arrangements", by Matthew Stamps (YaleNUS College)
Abstract. The Brieskorn and OrlikSolomon theorems state that the cohomology ring of the complement of a...
Location: Mathematics Department 
Jun 29 2016 GASC Seminar "Cohomology of Normalized BlowUps" by Manoj Kummini (Chennai Mathematical Institute)
Abstract: We prove some consequences of the vanishing of the higher cohomology of normalized blowups of threedimensional Noetherian local rings. These results are related to conjectures of Itoh...
Location: Department of Mathematics 
Abstract: "Graphings" are a natural generalization of finite graphs on probability measure spaces. These objects arise naturally as limits of finite graphs, as well as from the study of invariant...
Location: Department of Mathematics 
Abstract: A conjecture of Ronald Read predicts that the coefficients of the chromatic polynomial of a graph form a logconcave sequence for any graph. A related conjecture of Dominic Welsh...
Location: Department of Mathematics 
Apr 12 2016 GASC Seminar "Artin Groups and YokonumaHecke Algebras" by Ivan Marin (LAMFA, University of Picardy, Amiens)
Abstract: In this talk we introduce an extension C_W of the (Iwahori)Hecke algebra attached to an arbitrary Coxeter group W. We prove that C_W appears as a (1parameter family) of quotient(s) of...
Location: Department of Mathematics 
Abstract: Let A be an Artinian Gorenstein algebra over an infinite field of characteristic either 0 or greater than the socle degree of A. To every such algebra and a linear projection on its...
Location: Department of Mathematics 
Abstract: Mtheory compactifications on sevenmanifolds with G_2 holonomy give rise to N=1 theories in four dimensions, which may be relevant for particle physics. However, such compactifications...
Location: Department of Mathematics 
Abstract: When studying permutation groups, one typically looks first at the primitive permutation groups (transitive groups in which point stabilizers are maximal); in the finite case these...
Location: Department of Mathematics 
Abstract: The Hecke algebra H_n admits a categorification consisting of Soergel bimodules. As Khovanov showed, this category allows one to associate a triply graded vector space to every braid,...
Location: Department of Mathematics 
Abstract:
Let F be a field which is complete with respect to a discrete valuation and let G be an absolutely simple isotropic algebraic group defined over F. Then the group G(F) act...
Location: Department of Mathematics 
Abstract:
An abelian differential defines a flat metric with conical singularities such that the underlying Riemann surface can be realized as a plane polygon with suitable parallel edge...
Location: Department of Mathematics 
Feb 8 2016 GASC Seminar "How to Count Zeros Arithmetically?" by Jesse Kass (University of South Carolina)
Abstract: A celebrated result of EisenbudKimshaishvili–Levine computes the local degree of a smooth function f: R^n R^n with an isolated zero at the origin as the signature of the degree...
Location: Department of Mathematics 
Abstract: The theory of Bridgeland stability conditions has lead to deep results about the geometry of moduli spaces of sheaves on surfaces. One of the main obstacles to do the same on threefolds...
Location: Department of Mathematics 
Abstract: In this talk we show the existence of an integer for which the ArtinRees Lemma holds for all ideals and for all dth syzygies as sub modules of a free module. This work is a...
Location: Department of Mathematics 
Abstract: Understanding maps between algebraic varieties, schemes, or stacks is a difficult nonlinear problem. Lurie's Tannaka duality theorem allows one to identify maps between these objects...
Location: Department of Mathematics 
Abstract: A (v, k, ) BIBD (Balanced Incomplete Block Design) is an incidence structure of v points, a finite set of blocks of size k each, and an incidence relation between them such that every...
Location: Department of Mathematics
Monday, December 8, 2014
Speaker:  Javier Bracho (UNAM, visiting Northeastern) 
Title:  
Abstract: 
Monday, December 1, 2014
Speaker:  Nicholas Matteo (Northeastern University) 
Title:  
Abstract: 
Monday, November 24, 2014
Speaker:  Barbara Bolognese (Northeastern University) 
Title:  Generic strange duality on abelian surfaces 
Abstract:  Le Potier conjectured an unexpected duality between the complete linear series of certain natural divisors, called Theta divisors, on moduli spaces of sheaves on a surface. Such conjecture is widely known as Strange Duality conjecture. After the results of Marian and Oprea, who proved generic Strange Duality for sheaves on K3 surfaces, I will work in the setting of abelian surfaces. First, I will describe the geometric setup, including the moduli space, the Theta divisors and their relative version giving rise to the Verlinde sheaves. I will then present the duality in this setting and, time permitting, I will sketch its proof. This is joint work with A. Marian, D. Oprea and K. Yoshioka. 
Monday, November 17, 2014
Speaker:  Jarosław Buczyński (Polish Academy of Sciences) 
Title:  Apolarity and direct sum decompositions 
Abstract:  A polynomial is a direct sum if it can be written as a sum of two nonzero polynomials in some distinct sets of variables, up to a linear change of variables. We analyse criteria for a homogeneous polynomial to be decomposable as a direct sum, in terms of the apolar ideal of the polynomial. We prove that the apolar ideal of a polynomial of degree d strictly depending on all variables has a minimal generator of degree d if and only if it is a limit of direct sums. This is a joint work with Weronika Buczynska, Johannes Kleppe, and Zach Teitler. 
Monday, November 10, 2014
Speaker:  Alexandru Dimca (University of Nice and IAS Princeton) 
Title:  Jacobian relations and Hodge theory of singular hypersurfaces 
Abstract:  Let f in S=C[x_{0},…,x_{n}] be a homogeneous polynomial, and denote by f_{j} the partial derivative of f with respect to x_{j}, for j=0,…,n and J_{f} the Jacobian ideal spanned by them in S. Let V(f) be the projective hypersurface given by f=0 and F(f) the associated Milnor fiber given by f=1 in C^{n+1}. When V(f) is smooth, the partial derivatives f_{0},…,f_{n} form a regular sequence in S, the Hodge theory of the hypersurface V(f)(resp. of the Milnor fiber F(f)) was described by Ph. Griffiths (resp. by J. Steenbrink) in terms of the graded Jacobian (or Milnor) algebra S/J_{f}. In this talk we describe how these results change when V(f) acquires some isolated singularities, e.g. some nodes. The interplay between algebra and topology continues to function, but it becomes more involved. 
Monday, November 3, 2014
Speaker:  Idun Reiten (NTNU Trondheim) 
Title:  Support tautilting modules 
Abstract: 
Monday, October 27, 2014
Speaker:  Ramis Movassagh (Northeastern University) 
Title:  Eigenvalue attraction 
Abstract:  We prove that the complex conjugate eigenvalues of a real matrix attract in response to additive real randomness. Consider the time discretization 0< t_1< t_2<… and define a stochastic process by M(t_i + dt)=M(t_i)+dt*P(t_i), where M(0) is a fixed real matrix, t_i < dt and each is a real random whose entries are independent with zero mean and bounded we prove that any complex conjugate pair of eigenvalues of to prove we first construct a smooth family of stochastic processes such that M'(t)=P(t), that in the limit recover the original discrete process. We then explicitly write down the differential equations governing the motion of any eigenvalue. We derive formulas for the expectation value of the force and find that the force is inversely proportional to the distance of any c.c. pair and directly proportional to the 2norm squared of the corresponding left eigenvector. Therefore, c.c. pairs closest to the real axis, or those that are illconditioned, attract most strongly. We then prove that when the perturbation matrix is complex, there is no such force. A special limit of our results is applicable to the, often arising in application, small perturbations of a fixed matrix. We numerically illustrate the theory through various examples and discuss applications, including the HatanoNelson model. Time permitting we will discuss related results on the eigenpairs of Toeplitz matrices with singular FisherHartwig symbols and the aggregation and low density of the eigenvalues of real random matrices on and near the real axis respectively. 
Friday, October 24, 2014 (at 9:00am)
Speaker:  Larry Smith (University of Göttingen) 
Title:  Poincaré duality algebras, Milnor’s diagonal elements, and Macaulay duality 
Abstract:  In his famous lectures on characteristic classes John Milnor makes use of a cohomology class in H^*(M X M, Δ(M)) of a closed manifold M, where Δ: M → M X M is the diagonal embedding, to describe Poincaré duality and to define Stiefel–Whitney classes. In this talk I will make use of one of Milnor’s lemmas to give a purely algebraic definition of an element u ∈ A ⊗A, where A is a Poincaré duality algebra, that plays the same role. I will use this Milnor diagonal element to describe a Macaulay dual for the kernel of the multiplication map A ⊗ A → A, relate u to the dimension of A as a vector space, and to define Stiefel–Whitney classes if A supports an unstable Steenrod algebra action. Using this definition of Stiefel–Whitney classes I will show that Wu’s formula for the action of the Steenrod algebra on them holds, recovering a new proof of an old result of J.F. Adams. 
Monday, October 20, 2014
Speaker:  Claus Ringel (University of Bielefeld) 
Title:  The Auslander varieties for a wild algebra 
Abstract:  Let k be an algebraically closed field and A a finitedimensional (associative) kalgebra. Given an Amodule M, the set of all submodules of M with fixed dimension vector is called a quiver Grassmannian in M. If C and Y are Amodules, then we consider Hom(C,Y) as an E(C)module, where E(C) is the opposite of the endomorphism ring of C, and the Auslander varieties for A are the quiver Grassmannians in this E(C)module Hom(C,Y). In his seminal Philadelphia Notes (published in 1978), M. Auslander exhibited his theory of morphisms determined by modules. It is an important frame for describing the poset structure of the category of Amodules and this setting allows to interpret the Auslander varieties as describing factorizations of morphisms of Amodules. If the algebra A is (controlled) wild, then one knows that any projective variety can be realized as an Auslander variety. The aim of the lecture is to analyse sets of factorizations which can be used to realize arbitrary projective varieties. 
Monday, October 6, 2014
Speaker:  Karim Adiprasito (Hebrew University and IHES) 
Title:  Some remarks on the geometry of simplicial polytopes 
Abstract:  We show an interesting relation between the shape of a convex simplicial polytope, graph chordality and the Hard Lefschetz Theorem for projective toric varieties of simplicial polytopes. 
Friday, October 3, 2014 (at 10:30 am)
Speaker:  Eric Friedlander (University of Southern California) 
Title:  Invariants of representations using Jordan types 
Abstract:  This is will be a very informal discussion of how Julia Pevtsova and I have used the data of Jordan types of pnilpotent operators to give refined invariants (finer than support varieties) for modular representations. 
Wednesday, October 1, 2014
Speaker:  Justin Malestein (Hebrew University) 
Title:  Rigidity of Symmetric Frameworks 
Abstract:  Barjoint frameworks are structures made of fixedlength bars connected by universal joints with full rotational freedom. The allowed motions preserve the length and connectivity of the bars, and a framework is rigid if all the allowed motions extend to rigid body motions. For finite frameworks with generic geometry, rigidity is known to depend only on the graph that has as its edges the bars, and, in dimension 2, the generically rigid graphs are known exactly. In recent years, the question of extending this kind of combinatorial theory to infinite frameworks or finite frameworks with special geometry such as symmetry has received a lot of attention, motivated, in part, by applications in crystallography. In this talk, I will discuss both combinatorial and algebraic aspects of rigidity of barjoint frameworks with symmetry. In particular, I will present “Lamanlike” theorems for various kinds of symmetry in dimension 2; i.e. theorems which characterize generically rigid frameworks. The main focus of the talk will be on recent work of Theran and myself about rigidity and “ultrarigidity” of periodic frameworks (infinite frameworks in Euclidean space invariant under some lattice). 
Monday, September 29, 2014
Speaker:  Isabel Hubard (Instituto de Matematicas, UNAM, Mexico City) 
Title:  Chiral polytopes and alternating groups 
Abstract:  We review basic topics about abstract chiral polytopes and explain the difficulty of constructing such objects of high rank. Then a construction of chiral 4polytopes with automorphism groups isomorphic to alternating and symmetric groups is described. This is joint work with Marston Conder, Daniel Pellicer and Eugenia O’Reilly. 
Monday, September 22, 2014
Speaker:  Tony Iarrobino (Northeastern University) 
Title:  Jordan types of two commuting nilpotent matrices 
Abstract:  This work is joint with Leila Khatami, Bart Van Steirteghem and Rui Zhao. The similarity class of an n by n nilpotent matrix B over a field k is given by its Jordan type, which is the partition P of n that specifies the sizes of the Jordan blocks. The variety N(B) parametrizing nilpotent matrices that commute with B is irreducible, so there is a partition Q = Q(P) that is the generic Jordan type for matrices A in N(B). The partition Q(P) has parts that differ pairwise by at least two (we call it stable). P. Oblak proposed a recursive conjecture about the map P to Q(P); we (briefly) describe progress on it by P. Oblak, T. Kosir, L. Khatami, L.KhatamiI., D.I. Panyushev, and R. Basili.
Our focus, rather, is on the set S(Q) of partitions P having a given stable partition Q as generic nilpotent commuting orbit; and also on their loci Z(P), the set of A in N(B) having Jordan type P. We prove a Table Conjecture proposed by P. Oblak and R. Zhao: given Q = (u,ur) with u > r > 1 then S(Q) can be arranged in an (r1) by (ur) table T(Q) such that the partition P_{i.j} in the i,j entry has i + j parts. We conjecture with Mats Boij that the locus in N(B) of P_{i,j} is a complete intersection of i + j – 2 linear and quadratic equations. We propose a Box Conjecture for the set S(Q) for any stable Q.
Reference: ArXiv 1409.2192

Monday, September 8, 2014
Speaker:  Bruno Benedetti (Freie University Berlin, visiting Northeastern University) 
Title:  Diameter of graphs of polytopes and beyond 
Abstract:  The graph of every convex polytope is connected. Balinski proved it is even dconnected, where d is its dimension. But how far away can two vertices be, in a dpolytope with n facets? The “distance” is here just the number of edges you need to walk along to go from one to the other. Hirsch conjectured that the answer should be n–d, but the conjecture was disproved by Santos in 2010. I will explain a couple of recent positive results:(1) the Hirsch conjecture holds for all flag polytopes. The proof uses methods from metric geometry. This is joint work with Karim Adiprasito, arxiv:1303.3598.
(2) (if time permits) The StanleyReisner correspondence suggests an elementary way to define the dual graph also for algebraic varieties, basically by looking at “connectedness in codimension one”. Using an algebraic proof, we show that Balinski’s theorem can be extended to the generality of Gorenstein subspace arrangements. This is joint work with Matteo Varbaro, arXiv:1403.3241. 
Wednesday, April 16, 2014 at 10:30am
Speaker:  Thomas Church (Stanford University, visiting MIT) 
Title:  Uniform generators for the Johnson filtration 
Abstract: 
It’s wellknown that SL_{n}(Z) is generated by elementary matrices. The elementary matrices E_{ij} and E_{ji} are contained in the subgroup <E_{ij},E_{ji}> = SL_{2}(Z), so SL_{n}(Z) is generated by elements that are “supported” on some SL_{2}(Z) subgroup. Similarly, the mapping class group Mod_{g} is generated by Dehn twists supported on a genus1 subsurface.
The same question about subgroups is much harder! For congruence subgroups SL_{n}(Z,p), asking whether SL_{n}(Z,p) is generated by elementary matrices is essentially equivalent to the Congruence Subgroup Property. Johnson proved that the Torelli group Mod_{g}[1] is not generated by elements supported on genus1 subsurfaces. However, Birman–Powell proved that the Torelli group is generated by elements supported on genus2 subsurfaces.
I will give an overview of such “generated by elements of bounded support” results, and explain the ideas behind a new theorem: for every term Mod_{g}[k] of the Johnson filtration, there is a constant G_{k} so that Mod_{g}[k] is generated by elements supported on genusG_{k} subsurfaces. Joint work with Andrew Putman.

Monday, April 14, 2014
Speaker:  Jonathan Heckman (Harvard) 
Title:  What is a TBrane? 
Abstract: 
In string theory, the notion of the position of a particle in spacetime naturally generalizes to noncommuting matrices. Of particular interest are Tbranes, corresponding to the special case where such matrices are nilpotent. In this talk we focus on a particular manifestation of Tbranes in string theory described by solutions to Hitchin’s system with gauge group G. For G an ADE group, there is a wellestablished correspondence in the physics literature between nonsingular points of the Hitchin system moduli space and the deformation theory of a curve of ADE singularities. In singular limits such as those governed by Tbranes, the spectral equation for the Higgs field degenerates to zn=0, and the classical correspondence with deformation theory breaks down.
Using the theory of limiting mixed Hodge structures, we explain how to track this correspondence in such limits in the case of A_n type gauge groups / singularities. Time permitting, we discuss ongoing work on the extension of this work to situations other than Atype groups / singularities, highlighting the special case of Tbranes with no deformation moduli.
Based on joint work with L. Anderson, S. Katz, and L. Schaposnik.

Monday, April 7, 2014
Speaker:  Mark Mixer (Northeastern University) 
Title:  Transitive Permutation Groups as String CGroups of High Rank 
Abstract:  For any n ≥ 9, up to isomorphism and duality, there are exactly two string Cgroups of rank at least n2 that represent a transitive permutation group of degree n, in both cases the symmetric group Sym(n). In this talk I will extend this classification of high rank string Cgroups to include rank n − 3. The result is that for each n ≥ 9 there are seven string Cgroups of that rank that represent a transitive permutation group of degree n. Furthermore, the permutation group again is Sym(n) in each case. By completing this classification, I will construct all nondegenerate abstract regular polytopes of any rank d with automorphism group isomorphic to Sym(d+3).This is joint work with M.E. Fernandes and D. Leemans. 
Monday, March 31, 2014
Speaker:  Sarah Kitchen (University of Michigan) 
Title:  Generalized HarishChandra Modules 
Abstract:  HarishChandra modules, together with BeilinsonBernstein localization, are well known to allow the study of representations of real Lie groups from a complex algebraic perspective. These modules are simultaneously a representation of the complexification of the Lie algebra and maximal compact subgroup of the real group. Generalized HarishChandra modules weaken these requirements on the pair by taking any semisimple complex Lie algebra, and any reductive subalgebra. In this talk, I will explain new considerations that must be taken into account in order to localize these objects and a geometric approach to a conjecture of Penkov and Zuckerman, which categorifies a parameterization of some of the irreducible modules. 
Monday, March 24, 2014
Speaker:  Max Wakefield (US Naval Academy) 
Title:  Limits of rational models of kequal arrangements 
Abstract:  Braid arrangements arise through many different fields of mathematics as configurations spaces, as reflecting hyperplanes of type A Coxeter groups, as realizations of partition lattices, and as a method to compute chromatic polynomials of graphs. We define a monoidal structure on a set of generalizations (kequal arrangements) of braid arrangements and study some limits of these arrangements and their rational models. Summing these limits together we present a kind of classifying algebra for the rational homotopy models of the complements of limits of kequal arrangements. This algebra has a rich combinatorial structure. We will discuss some of its properties and applications. 
Monday, March 17, 2014
Speaker:  Ivan Losev (Northeastern University) 
Title:  Procesi bundles on Hilb^n(C^2) 
Abstract:  A Procesi bundle is a vector bundle on the Hilbert scheme of n points on the plane. It was first constructed by Haiman who used it to prove the Schur positivity for Macdonald polynomials. This bundle also provides a derived McKay equivalence for the Hilbert scheme. I will basically take the latter for an axiomatic description of a Procesi bundle. I will show that there are exactly two bundles with these properties: Haiman’s and its dual. Time permitting I will also discuss an extension of these results to other symplectic resolutions and a relation between the Procesi bundles and the tautological bundle conjectured by Haiman. The proofs are based on the study of symplectic reflection algebras. 
Monday, March 10, 2014
Speaker:  Julia Pevtsova (University of Washington) 
Title:  Modules of Constant Jordan Type and Their Generalizations 
Abstract:  Let G be a finite group (scheme) with the group algebra kG for a field k of positive characteristic p. A module of constant Jordan type over G is a module that exhibits the same behavior when restricted to various subalgebras of kG isomorphic to k[t]/t^p. I’ll discuss properties of these modules and their generalizations, some open problems and recent progress towards them, and the connections between modules of constant Jordan type and the geometry of the projective variety Proj H^*(G,k). For many of my examples, I’ll concentrate on the case of an elementary abelian pgroup, for which the geometric connections lead to a correspondence between modules of constant Jordan type and vector bundles on projective spaces (and between generalized modules of constant Jordan type and bundles on Grassmannians).This is joint work with J. Carlson and E. Friedlander. 
Monday, February 24, 2014
Speaker:  Anand Patel (Boston College) 
Title:  The Maroni Theory of Hurwitz Space 
Abstract:  In this talk I will discuss what I call the “Maroni theory” of the Hurwitz spaces parametrizing branched covers of a fixed curve. The goal is to understand the geometry of a decomposition of Hurwitz space into the union of certain special subvarieties, the socalled Maroni loci. This decomposition seems to play an important role in the solution to various problems. I will discuss some of these problems. No prior familiarity of Hurwitz spaces will be assumed. 
Thursday, December 13, 2013
Speaker:  Diane Maclagan (University of Warwick) 
Title:  The Cox ring of wonderful compactifications 
Abstract:  The wonderful model Y~ of a hyperplane arrangement complement Y is a smooth compactification of Y with normal crossing boundary introduced by DeConcini and Procesi. Examples include the blowup of P^{2} at any number of points, and the moduli space M‾_{0,n}. I will introduce these varieties, and describe an invariant ring description of their Cox rings generalizing the DoranGiansiracusa description for M‾_{0,n}, plus an associated power ideal description of the graded pieces of the Cox rings that generalizes the EmsalemIarrobino description for blowups of P^{2}. This is joint work with Florian Block. 