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64
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Cited by 38 (15 self)
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Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
Modular compactifications of the space of pointed elliptic curves I
"... Abstract. We prove that the moduli spaces of npointed mstable curves introduced in our previous paper have projective coarse moduli. We use the resulting spaces to run an analogue of Hassett’s log minimal model program for M1,n. ..."
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Cited by 16 (4 self)
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Abstract. We prove that the moduli spaces of npointed mstable curves introduced in our previous paper have projective coarse moduli. We use the resulting spaces to run an analogue of Hassett’s log minimal model program for M1,n.
On Lines and Joints
, 2009
"... Let L be a set of n lines in R d, for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplifica ..."
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Cited by 13 (2 self)
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Let L be a set of n lines in R d, for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz [9], and of the followup simpler proof of Elekes et al. [6]. Some extensions, e.g., to the case of joints of algebraic curves, are also presented.
Conca: New free divisors from old
"... Abstract. We present several methods to construct or identify families of free divisors such as those annihilated by many Euler vector fields, including binomial free divisors, or divisors with triangular discriminant matrix. We show how to create families of quasihomogeneous free divisors through t ..."
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Cited by 8 (1 self)
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Abstract. We present several methods to construct or identify families of free divisors such as those annihilated by many Euler vector fields, including binomial free divisors, or divisors with triangular discriminant matrix. We show how to create families of quasihomogeneous free divisors through the chain rule or by extending them into the tangent bundle. We also discuss whether general divisors can be extended to free ones by adding components and
Critical Points and Gröbner Bases: the Unmixed Case
, 2012
"... We consider the problem of computing critical points of the restriction of a polynomial map to an algebraic variety. This is of first importance since the global minimum of such a map is reached at a critical point. Thus, these points appear naturally in nonconvex polynomial optimization which occu ..."
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Cited by 8 (5 self)
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We consider the problem of computing critical points of the restriction of a polynomial map to an algebraic variety. This is of first importance since the global minimum of such a map is reached at a critical point. Thus, these points appear naturally in nonconvex polynomial optimization which occurs in a wide range of scientific applications (control theory, chemistry, economics,...). Critical points also play a central role in recent algorithms ofeffectiverealalgebraicgeometry. Experimentally, it has been observed that Gröbner basis algorithms are efficient to compute such points. Therefore, recent software based on the socalled Critical Point Method are built on Gröbner bases engines. Let f1,...,fp be polynomials in Q[x1,...,xn] of degree D, V ⊂ C n be their complex variety and π1 be the projection map (x1,...,xn) ↦ → x1. Thecriticalpointsoftherestrictionofπ1to V are defined by the vanishing of f1,...,fp and some maximal minors of the Jacobian matrix associated to f1,...,fp. Suchasystemisalgebraicallystructured:theidealitgenerates is the sum of a determinantal ideal and the ideal generated by f1,...,fp. We provide the first complexity estimates on the computation of Gröbner bases of such systems defining critical points. We prove that under genericity assumptions on f1,...,fp, thecomplexityis polynomial in the generic number of critical points, i.e. D p (D − 1) n−p () n−1.Moreparticularly,inthe p−1 quadratic case D =2,thecomplexityofsuchaGröbnerbasiscomputationispolynomial in the number of variables n and exponential in p. We also give experimental evidence supporting these theoretical results.
On the Complexity of the Generalized MinRank Problem
, 2013
"... We study the complexity of solving the generalized MinRank problem, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most r. A natural algebraic representation of this problem gives rise to a determinantal ideal: the ideal generated by all minors of size r + 1 ..."
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Cited by 7 (4 self)
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We study the complexity of solving the generalized MinRank problem, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most r. A natural algebraic representation of this problem gives rise to a determinantal ideal: the ideal generated by all minors of size r + 1 of the matrix. We give new complexity bounds for solving this problem using Gröbner bases algorithms under genericity assumptions on the input matrix. In particular, these complexity bounds allow us to identify families of generalized MinRank problems for which the arithmetic complexity of the solving process is polynomial in the number of solutions. We also provide an algorithm to compute a rational parametrization of the variety of a 0dimensional and radical system of bidegree (D,1). We show that its complexity can be bounded by using the complexity bounds for the generalized MinRank problem.
Deformation theory from the point of view of fibered categories
"... Abstract. We give an exposition of the formal aspects of deformation theory in the language of fibered categories, instead of the more traditional one of functors. The main concepts are that of tangent space to a deformation problem, obstruction theory, versal and universal formal deformations. We ..."
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Cited by 7 (0 self)
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Abstract. We give an exposition of the formal aspects of deformation theory in the language of fibered categories, instead of the more traditional one of functors. The main concepts are that of tangent space to a deformation problem, obstruction theory, versal and universal formal deformations. We include proofs of two key results: a version of Schlessinger’s Theorem in this context, and the Ran–Kawamata vanishing theorem for obstructions. We accompany this with a detailed analysis of three important cases: smooth varieties, local complete intersection subschemes and coherent sheaves.
Associated forms and hypersurface singularities: the binary case, preprint
"... Abstract. In the recent articles [EI] and [AI], it was conjectured that all rational GLninvariant functions of forms of degree d ≥ 3 on Cn can be extracted, in a canonical way, from those of forms of degree n(d−2) by means of assigning every form with nonvanishing discriminant the socalled associa ..."
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Cited by 3 (1 self)
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Abstract. In the recent articles [EI] and [AI], it was conjectured that all rational GLninvariant functions of forms of degree d ≥ 3 on Cn can be extracted, in a canonical way, from those of forms of degree n(d−2) by means of assigning every form with nonvanishing discriminant the socalled associated form. While this surprising statement is interesting from the point of view of classical invariant theory, its original motivation was the reconstruction problem for isolated hypersurface singularities, which is the problem of finding a constructive proof of the wellknown MatherYau theorem. The conjecture was confirmed in [EI] for binary forms of degree d ≤ 6 as well as ternary cubics. Furthermore, a weaker version of it was settled in [AI] for arbitrary n and d. In the present paper, we focus on the case n = 2 and establish the conjecture, in a rather explicit way, for binary forms of an arbitrary degree. This result allows one to extract a complete system of biholomorphic invariants of homogeneous plane curve singularities from their Milnor algebras. 1.