Blowups and Fibers of Morphisms

Kustin, Andrew R.; Polini, Claudia; Ulrich, Bernd

doi:10.1017/nmj.2016.34

Cited by 21 publications

(15 citation statements)

References 50 publications

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“…If R mR is a DVR then λ (m i G mG /m i+1 G mG ) = 1 for every 0 i < t. Since e mR mR = 1 it follows from [34,Theorem 5.3] that Quot F = Quot K[A t ] , i.e., every minimal generator of m t is a fraction of elements of F. The latter is equivalent to the existence of an integer n 0 and an element g ∈ I n \ mI n such that gm t ⊂ I n+1 , i.e., m t G mG = 0. Hence the equality in (ii) holds.…”

Section: J-multiplicity and Degree Of The Fibermentioning

confidence: 97%

Multiplicities of classical varieties

Jeffries

Montaño

Varbaro

2015

Proceedings of the London Mathematical Society

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The j‐multiplicity plays an important role in the intersection theory of Stückrad–Vogel cycles, while recent developments confirm the connections between the ϵ‐multiplicity and equisingularity theory. In this paper, we establish, under some constraints, a relationship between the j‐multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the j‐multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the j‐ and ϵ‐multiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory.

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Section: J-multiplicity and Degree Of The Fibermentioning

confidence: 97%

Multiplicities of classical varieties

Jeffries

Montaño

Varbaro

2015

Proceedings of the London Mathematical Society

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“…Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of a rational map, which is called the base ideal of the map. This idea goes back to [36] and since then a large amount of papers has blossomed in this direction (see, for example, [5,17,19,28,29,43,49,50,54]).…”

Section: Introductionmentioning

confidence: 99%

Degree and birationality of multi‐graded rational maps

Busé

Cid-Ruiz

DʼAndrea

2020

Proc. London Math. Soc.

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We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of the rational map. A key ingredient is a new algebra that we call the saturated special fiber ring, which turns out to be a fundamental tool to analyze the degree of a rational map. We also provide a very effective birationality criterion and a complete description of the equations of the associated Rees algebra of a particular class of plane rational maps.

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“…We first argue the third equivalence in item (1.b). Again, as in the proof of Lemma 2.10, one obtains a regular sequence u, v of forms of degree d/(deg C ) so that I = I 2 (ϕ ′ )R for some 3 × 2 matrix ϕ ′ whose entries in position i, j are homogeneous polynomials of degree d j (deg C )/d in the variables u, v (see [25]). Thus one reduces to the case of a birational parametrization.…”

Section: Proofmentioning

confidence: 97%

“…Let r be the degree of the field extension [Quot(k[R d ]) : Quot(k[I d ])], as described in Remark 2.9. Then there exists a regular sequence u, v in R r so that I = I 2 (ϕ ′ )R for some 3 × 2 matrix ϕ ′ whose entries are homogeneous polynomials of degree (deg C )/2 = d/2r in the variables u, v (see [25]). The signed maximal minors of ϕ ′ provide a birational parametrization of the same curve C .…”

Section: Proofmentioning

confidence: 99%

The bi-graded structure of symmetric algebras with applications to Rees rings

2017

Self Cite

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Consider a rational projective plane curve C parameterized by three homogeneous forms of the same degree in the polynomial ring R = k [x, y] over a field k. The ideal I generated by these forms is presented by a homogeneous 3 × 2 matrix ϕ with column degrees d 1 ≤ d 2 . The Rees algebra R = R[It] of I is the bi-homogeneous coordinate ring of the graph of the parameterization of C ; and accordingly, there is a dictionary that translates between the singularities of C and algebraic properties of the ring R and its defining ideal. Finding the defining equations of Rees rings is a classical problem in elimination theory that amounts to determining the kernel A of the natural map from the symmetric algebra Sym(I) onto R . The ideal A ≥d 2 −1 , which is an approximation of A, can be obtained using linkage. We exploit the bi-graded structure of Sym(I) in order to describe the structure of an improved approximation A ≥d 1 −1 when d 1 < d 2 and ϕ has a generalized zero in its first column. (The latter condition is equivalent to assuming that C has a singularity of multiplicity d 2 .) In particular, we give the bi-degrees of a minimal bi-homogeneous generating set for this ideal. When 2 = d 1 < d 2 and ϕ has a generalized zero in its first column, then we record explicit generators for A. When d 1 = d 2 , we provide a translation between the bi-degrees of a bi-homogeneous minimal generating set for A d 1 −2 and the number of singularities of multiplicity d 1 that are on or infinitely near C . We conclude with a table that translates between the bi-degrees of a bi-homogeneous minimal generating set for A and the configuration of singularities of C when the curve C has degree six.

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Blowups and Fibers of Morphisms

Abstract: Our object of study is a rational map Ψ : P s−1 k

Cited by 21 publications

References 50 publications

Multiplicities of classical varieties

Multiplicities of classical varieties

Degree and birationality of multi‐graded rational maps

The bi-graded structure of symmetric algebras with applications to Rees rings

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