Blowup algebras of rational normal scrolls

Sammartano, Alessio

doi:10.1090/tran/7689

Cited by 7 publications

(19 citation statements)

References 29 publications

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“…(I) We first show that the equality (34) holds when c = r + d + 1 via the lex type ordering. Whence, the right-hand side of ( 34) is given by lex r,r+d+1,d , which surely satisfies lex r,r+d+1,d ≤ revlex r,r+d+1,d by (32). (II) Next, we show that (34) holds when c = 2r + d − 1 via the revlex type ordering.…”

Section: 3mentioning

confidence: 90%

“…Proof. The condition (C1) is obviously necessary in view of (32). Now, we go through the iterated mapping cone process.…”

Section: 3mentioning

confidence: 99%

“…This generator surely cannot be canceled by a corner generator. Hence the maximum at the right-hand side of (32) is exactly (N − 1) − (N − 1)/r + 1 = (N − 1) − (N − 1)/r .…”

mentioning

confidence: 94%

“…The implicitization problem for the blow-up algebras of the rational normal scrolls is very difficult. It was until recently solved by Sammartano in full generality in [32]. With this important progress, he was then able to prove that the Rees algebra and the fiber cones of the rational normal scrolls are Koszul algebras.…”

Section: Introductionmentioning

confidence: 99%

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Blow-up algebras of secant varieties of rational normal scrolls

Lin¹,

Shen²

2021

Preprint

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In this paper, we are mainly concerned with the blow-up algebras of the secant varieties of balanced rational normal scrolls. In the first part, we give implicit defining equations of their associated Rees algebras and fiber cones. Consequently, we can tell that the fiber cones are Cohen-Macaulay normal domains. Meanwhile, these fiber cones have rational singularities in characteristic zero, and are F -rational in positive characteristic. The Gorensteinness of the fiber cones can also be characterized. In the second part, we compute the Castelnuovo-Mumford regularities and a-invariants of the fiber cones. We also present the reduction numbers of the ideals defined by the secant varieties.

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Section: 3mentioning

confidence: 90%

“…Proof. The condition (C1) is obviously necessary in view of (32). Now, we go through the iterated mapping cone process.…”

Section: 3mentioning

confidence: 99%

“…This generator surely cannot be canceled by a corner generator. Hence the maximum at the right-hand side of (32) is exactly (N − 1) − (N − 1)/r + 1 = (N − 1) − (N − 1)/r .…”

mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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Blow-up algebras of secant varieties of rational normal scrolls

Lin¹,

Shen²

2021

Preprint

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“…The ideal of maximal minors of a generic matrix was shown by Conca and his coauthors in [5]. The rational normal scrolls associated with a 2 × n matrix were shown by Sammartano in [18]. Very recently, the two-minors of a generic 3 × n matrix was resolved by Huang and her coauthors in [15].…”

Section: Introductionmentioning

confidence: 99%

Rees Algebras of Closed Determinantal Facet Ideals

Almousa¹,

Lin²,

Liske³

2020

Preprint

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Using SAGBI basis techniques, we find Gröbner bases for the presentation ideals of the Rees algebra and special fiber ring of a closed determinantal facet ideal. In particular, we show that closed determinantal facet ideals are of fiber type and their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains, and have rational singularities.

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