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332
The Intrinsic Normal Cone
 INVENT. MATH
, 1997
"... We suggest a construction of virtual fundamental classes of certain types of moduli spaces. ..."
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Cited by 347 (9 self)
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We suggest a construction of virtual fundamental classes of certain types of moduli spaces.
A Survey on the Model Theory of Difference Fields
, 2000
"... We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the various concepts of rank, the dichotomy theorems, and, as an application, the Manin–Mumford conject ..."
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Cited by 101 (16 self)
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We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the various concepts of rank, the dichotomy theorems, and, as an application, the Manin–Mumford conjecture over a number field. We conclude with some other applications.
Vector bundles over elliptic fibrations
"... Let π: Z → B be an elliptic fibration with a section. The goal of this paper is to study holomorphic vector bundles over Z. We are mainly concerned with vector bundles V with trivial determinant, or more generally such that detV has trivial restriction to each fiber, so that detV is the pullback of ..."
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Cited by 87 (4 self)
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Let π: Z → B be an elliptic fibration with a section. The goal of this paper is to study holomorphic vector bundles over Z. We are mainly concerned with vector bundles V with trivial determinant, or more generally such that detV has trivial restriction to each fiber, so that detV is the pullback of a line bundle on B. (The
A conjectural generating function for numbers of curves on surfaces
, 1997
"... I give a conjectural generating function for the numbers of δnodal curves in a linear system of dimension δ on an algebraic surface. It reproduces the results of Vainsencher [V2] for the case δ ≤ 6 and KleimanPiene [KP] for the case δ ≤ 8. The numbers of curves are expressed in terms of five uni ..."
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Cited by 65 (3 self)
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I give a conjectural generating function for the numbers of δnodal curves in a linear system of dimension δ on an algebraic surface. It reproduces the results of Vainsencher [V2] for the case δ ≤ 6 and KleimanPiene [KP] for the case δ ≤ 8. The numbers of curves are expressed in terms of five universal power series, three of which I give explicitly as quasimodular forms. This gives in particular the numbers of curves of arbitrary genus on a K3 surface and an abelian surface in terms of quasimodular forms, generalizing the formula of YauZaslow for rational curves on K3 surfaces. The coefficients of the other two power series can be determined by comparing with the recursive formulas of CaporasoHarris for the Severi degrees in P2. We verify the conjecture for genus 2 curves on an abelian surface. We also discuss a link of this problem with Hilbert schemes of points.
When polynomial equation systems can be "solved" fast?
 IN PROC. 11TH INTERNATIONAL SYMPOSIUM APPLIED ALGEBRA, ALGEBRAIC ALGORITHMS AND ERRORCORRECTING CODES, AAECC11
, 1995
"... We present a new method for solving symbolically zerodimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of an alternative data structure: arithmetic networks and straightline programs with FOR gates. For sequential time complexity measu ..."
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Cited by 65 (19 self)
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We present a new method for solving symbolically zerodimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of an alternative data structure: arithmetic networks and straightline programs with FOR gates. For sequential time complexity measured by the size of these networks we obtain the following result: it is possible to solve any affine or toric zerodimensional equation system in nonuniform sequential time which is polynomial in the length of the input description and the &quot;geometric degree &quot; of the equation system. Here, the input is thought to be given by a straightline program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). Geometric degree has to be adequately defined. It is always bounded by the algebraiccombinatoric &quot;B'ezout number &quot; of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometric degree is much smaller than
Straightline programs in geometric elimination theory
 J. Pure Appl. Algebra
, 1998
"... Dedicated to Volker Strassen for his work on complexity We present a new method for solving symbolically zero–dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures: arithmetic networks and straight–line prog ..."
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Cited by 64 (16 self)
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Dedicated to Volker Strassen for his work on complexity We present a new method for solving symbolically zero–dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures: arithmetic networks and straight–line programs. For sequential time complexity measured by network size we obtain the following result: it is possible to solve any affine or toric zero–dimensional equation system in non–uniform sequential time which is polynomial in the length of the input description and the “geometric degree ” of the equation system. Here, the input is thought to be given by a straight–line program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). The geometric degree of the input system has to be adequately defined. It is always bounded by the algebraic–combinatoric “Bézout number ” of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometric
Algebraic Geometry
, 2002
"... Notes for a class taught at the University of Kaiserslautern 2002/2003 ..."
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Cited by 51 (0 self)
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Notes for a class taught at the University of Kaiserslautern 2002/2003
Algebraic cobordism revisited
"... Abstract. We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provid ..."
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Cited by 50 (7 self)
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Abstract. We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative DonaldsonThomas theory. We use double point cobordism to prove all the degree 0 conjectures in DonaldsonThomas theory: absolute, relative, and equivariant. 0.1. Overview. A first idea for defining cobordism in algebraic geometry is to impose the relation