Minimal free resolutions of the tangent cones for Gorenstein monomial curves

Abstract We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We partition the moduli $\mathcal{M}$ according to the Hilbert function H and classify all possible Hilbert functions H of such bundles. For each H, we describe a stratification of $\mathcal{M}_H$ by quotients of rational varieties. We show that the closed strata form a graded lattice given by the Betti numbers.

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Complete Intersection Monomial Curves and the Cohen—Macaulayness of Their Tangent Cones

Algebra Colloquium ◽

10.1142/s1005386719000464 ◽

2019 ◽

Vol 26 (04) ◽

pp. 629-642

Author(s):

Anargyros Katsabekis

Keyword(s):

Complete Intersection ◽

Tangent Cone ◽

Affine Space ◽

Intersection Property ◽

Tangent Cones ◽

Monomial Curves ◽

Monomial Curve

Let C(n) be a complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the complete intersection property of the monomial curve C(n + wv), where w > 0 is an integer and v ∈ ℕ4. In addition, we investigate the Cohen–Macaulayness of the tangent cone of C(n + wv).

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Determinantal ideals without minimal free resolutions

Nagoya Mathematical Journal ◽

10.1017/s0027763000003081 ◽

1990 ◽

Vol 118 ◽

pp. 203-216 ◽

Cited By ~ 16

Author(s):

Mitsuyasu Hashimoto

Keyword(s):

Free Resolution ◽

Unit Element ◽

Free Resolutions ◽

Minimal Free Resolution ◽

Determinantal Ideals ◽

Arbitrary Base ◽

Generic Matrix ◽

Base Ring ◽

Minimal Free Resolutions ◽

The Ideal

Let R be a Noetherian commutative ring with, unit element, and Xij be variables with 1 ≤ i ≤ m and 1 ≤ j ≤ n. Let S = R[xij] be the polynomial ring over R, and It be the ideal in S, generated by the t × t minors of the generic matrix (xij) ∈ Mm, n(S). For many years there has been considerable interest in finding a minimal free resolution of S/It, over arbitrary base ring R. If we have a minimal free resolution P. over R = Z, the ring of integers, then R′ ⊗z P. is a resolution of S/It over the base ring R′.

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LIFTINGS OF A MONOMIAL CURVE

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972718000400 ◽

2018 ◽

Vol 98 (2) ◽

pp. 230-238

Author(s):

MESUT ŞAHİN

Keyword(s):

Tangent Cone ◽

Tangent Cones ◽

Monomial Curves ◽

Original Curve ◽

Monomial Curve

We study an operation, that we call lifting, creating nonisomorphic monomial curves from a single monomial curve. Our main result says that all but finitely many liftings of a monomial curve have Cohen–Macaulay tangent cones even if the tangent cone of the original curve is not Cohen–Macaulay. This implies that the Betti sequence of the tangent cone is eventually constant under this operation. Moreover, all liftings have Cohen–Macaulay tangent cones when the original monomial curve has a Cohen–Macaulay tangent cone. In this case, all the Betti sequences are just the Betti sequence of the original curve.

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