High-order vanishing ideals at n+3 points of P

D'Cruz, Clare; Iarrobino, Anthony

doi:10.1016/s0022-4049(99)00125-5

Cited by 2 publications

(21 citation statements)

References 12 publications

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“…We resolve a problem left open by Buczyńska and Wiśniewski in [6], by showing that the phylogenetic varieties in [6,32] arise as flat limits of the Cox-Nagata rings constructed by Castravet and Tevelev in [8]. We also prove a conjecture of D'Cruz and Iarrobino [10] on Hilbert functions of fat points. A key tool is an interpretation of the Verlinde formula [23] suggested to us by Jenia Tevelev.…”

Section: Example 12 (D = 2) For Pairwise Linearly Independent Binarmentioning

confidence: 80%

“…Hilbert functions of ideals generated by powers of n general linear forms in n − 2 unknowns were studied by D'Cruz and Iarrobino in [10]. The previous two theorems establish both parts of their main conjecture on page 77 of [10].…”

Section: Theorem 72 (Verlinde Formulamentioning

confidence: 88%

“…The following classical fact relates our problem to the study of ideals of fat points [10]. We learned Lemma 2.3 from Emsalem and Iarrobino [14, Theorem 1].…”

Section: Cox-nagata Ringsmentioning

confidence: 99%

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Sagbi bases of Cox–Nagata rings

Sturmfels

2010

J. Eur. Math. Soc.

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Abstract. We degenerate Cox-Nagata rings to toric algebras by means of sagbi bases induced by configurations over the rational function field. For del Pezzo surfaces, this degeneration implies the Batyrev-Popov conjecture that these rings are presented by ideals of quadrics. For the blow-up of projective n-space at n + 3 points, sagbi bases of Cox-Nagata rings establish a link between the Verlinde formula and phylogenetic algebraic geometry, and we use this to answer questions due to D'Cruz-Iarrobino and Buczyńska-Wiśniewski. Inspired by the zonotopal algebras of Holtz and Ron, our study emphasizes explicit computations, and offers a new approach to Hilbert functions of fat points.

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Section: Example 12 (D = 2) For Pairwise Linearly Independent Binarmentioning

confidence: 80%

Section: Theorem 72 (Verlinde Formulamentioning

confidence: 88%

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Sagbi bases of Cox–Nagata rings

Sturmfels

2010

J. Eur. Math. Soc.

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“…since both Syz(I) and t H 0 (S(I)(t)) are direct sums of kernels of the same maps on global sections. From equation (5) we also see that…”

Section: Taking Cohomology Shows Thatmentioning

confidence: 86%

“…We also have Syz(I ) = Syz(I) ⊗ S and thus S(I ) = S(I)| L = S(I) ⊗ S . Indeed, tensoring equation (5) by S yields the sequence…”

Section: Taking Cohomology Shows Thatmentioning

confidence: 99%

Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property

Harbourne

Schenck

Seceleanu

2011

Journal of the London Mathematical Society

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In a recently published paper [Trans. Amer. Math. Soc. 363 (2011) 229-257], Migliore, Miró-Roig and Nagel show that the weak Lefschetz property (WLP) can fail for an ideal I ⊆ K[x 1, . . . , x4] generated by powers of linear forms. This is in contrast to the analogous situation in K[x 1, x2, x3], where WLP always holds [H. Schenck and A. Seceleanu, Proc. Amer. Math. Soc. 138 (2010) 2335-2339]. We use the inverse system dictionary to connect I to an ideal of fat points, and show that the failure of WLP for powers of linear forms is connected to the geometry of the associated fat point scheme. Recent results of Sturmfels and Xu [J. Eur. Math. Soc. 12 (2010) 429-459] allow us to relate the WLP to Gelfand-Tsetlin patterns.

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High-order vanishing ideals at n+3 points of P

Cited by 2 publications

References 12 publications

Sagbi bases of Cox–Nagata rings

Sagbi bases of Cox–Nagata rings

Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property

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