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67
Murphy’s law in algebraic geometry: Badlybehaved deformation spaces
 Invent. Math
"... ABSTRACT. We consider the question: “How bad can the deformation space of an object be? ” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible. ” We show this for a number of important moduli spaces. More precisely, every singularit ..."
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Cited by 40 (4 self)
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ABSTRACT. We consider the question: “How bad can the deformation space of an object be? ” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible. ” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over Z (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective generaltype surfaces (or higherdimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces have very ample canonical bundle, the stable sheaves are torsionfree of rank 1, the singularities are normal and CohenMacaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of wellbehaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given nonreduced behavior along various associated subschemes. Similarly one can give a surface over Fp that lifts to Z/p 7 but not Z/p 8. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mnëv’s Universality Theorem.
Numerical Stability of Algorithms for Line Arrangements
 In Proc. 7th Annu. ACM Sympos. Comput. Geom
, 1991
"... We analyze the behavior of two line arrangement algorithms, a sweepline algorithm and an incremental algorithm, in approximate arithmetic. The algorithms have running times O(n 2 log n) and O(n 2 ) respectively. We show that each of these algorithms can be implemented to have O(nffl) relative e ..."
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Cited by 29 (6 self)
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We analyze the behavior of two line arrangement algorithms, a sweepline algorithm and an incremental algorithm, in approximate arithmetic. The algorithms have running times O(n 2 log n) and O(n 2 ) respectively. We show that each of these algorithms can be implemented to have O(nffl) relative error. This means that each algorithm produces an arrangement realized by a set of pseudolines so that each pseudoline differs from the corresponding line relatively by at most O(nffl). We also show that there is a line arrangement algorithm with O(n 2 log n) running time and O(ffl) relative error. 1 Introduction We analyze the behavior of line arrangement algorithms in approximate arithmetic. Approximate arithmetic is a set of arithmetic operations defined on the real numbers that make relative error ffl; this models floating point arithmetic. The input to a line arrangement algorithm is a set of n lines specified by real number coefficients. The output is a "combinatorial arrangement", ...
Double precision geometry: A general technique for calculating line and segment intersections using rounded arithmetic
 In IEEE Annual Symposium on Foundations of Computer Science
, 1989
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Finding Compact Coordinate Representations for Polygons and Polyhedra
 IBM Journal of Research and Development
, 1989
"... Practical solid modeling systems are plagued by numerical problems that arise from using floatingpoint arithmetic. For example, polyhedral solids are often represented by a combination of geometric and combinatorial information. The geometric information might consist of explicit plane equations, wi ..."
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Cited by 21 (4 self)
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Practical solid modeling systems are plagued by numerical problems that arise from using floatingpoint arithmetic. For example, polyhedral solids are often represented by a combination of geometric and combinatorial information. The geometric information might consist of explicit plane equations, with floatingpoint coefficients; the combinatorial information might consist of face, edge, and vertex adjacencies and orientations, with edges defined by faceface adjacencies and vertices by edgeedge adjacencies. Problems arise when numerical error in geometric operations causes the geometric information to become inconsistent with the combinatorial information. These problems could be avoided by using exact arithmetic instead of floatingpoint arithmetic. However, some operations, like rotation, increase the number of bits required to represent the plane equation coefficients. Since the execution time of exact arithmetic operators increases with the number of bits in the operands, the inc...
RIGIDITY AND POLYNOMIAL INVARIANTS OF CONVEX POLYTOPES
, 2004
"... We present an algebraic approach to the classical problem of constructing a simplicial convex polytope given its planar triangulation and lengths of its edges. We introduce polynomial invariants of a polytope and show that they satisfy polynomial relations in terms of squares of edge lengths. We obt ..."
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Cited by 19 (5 self)
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We present an algebraic approach to the classical problem of constructing a simplicial convex polytope given its planar triangulation and lengths of its edges. We introduce polynomial invariants of a polytope and show that they satisfy polynomial relations in terms of squares of edge lengths. We obtain sharp upper and lower bounds on the degree of these polynomial relations. In a special case of regular bipyramid we obtain explicit formulae for some of these relations. We conclude with a proof of Robbins Conjecture [R2] on the degree of generalized Heron polynomials.
Robust Polygon Modeling
 COMPUTERAIDED DESIGN
, 1993
"... The article provides a set of algorithms for performing set operations on polygonal regions in the plane using standard floating point arithmetic. The algorithms are robust, guaranteeing both topological consistency and numerical accuracy. Each polygon edge is modeled as an implicit or explicit poly ..."
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Cited by 17 (4 self)
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The article provides a set of algorithms for performing set operations on polygonal regions in the plane using standard floating point arithmetic. The algorithms are robust, guaranteeing both topological consistency and numerical accuracy. Each polygon edge is modeled as an implicit or explicit polygonal curve which stays within some distance fi of the original line segment. If the curve is implicit, fi is bounded by a small multiple of the rounding unit. If the curves are explicit, the bound on fi may grow with the number of curves. One can mix implicit and explicit representations to suit the application.
Realization Spaces of 4Polytopes are Universal
 BULL. AMER. MATH. SOC
, 1995
"... Let P be a ddimensional polytope. The realization space of P is the space of all polytopes P # that are combinatorially equivalent to P , modulo affine transformations. We report ..."
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Cited by 16 (4 self)
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Let P be a ddimensional polytope. The realization space of P is the space of all polytopes P # that are combinatorially equivalent to P , modulo affine transformations. We report
Some Algorithmic Problems in Polytope Theory
 IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS
, 2003
"... Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact... ..."
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Cited by 14 (0 self)
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Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...