Syzygies in Hilbert schemes of complete intersections

Caviglia, Giulio; Sammartano, Alessio

doi:10.48550/arxiv.1903.08770

Cited by 3 publications

(4 citation statements)

References 25 publications

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“…In other words, Epdq " K `mr`1 for an ideal K generated by a lexicographic initial segment of monomials of degree r, for some r. It is clear that Epdq " m r when d " `r`2 3 ˘. More generally, the ideal Epdq behaves in many respects in the same way as the powers of m, attaining for every d extremal number of generators, syzygies, socle monomials, and more, see [CS21,V94]. Conjecture 2 stated that Epdq also attains extremal tangent space dimension, for every d. Sturmfels [S00] disproved it by exhibiting counterexamples for d " 8, 16.…”

Section: B -Imentioning

confidence: 99%

On the tangent space to the Hilbert scheme of points in 𝐏³

Ramkumar

Sammartano

2022

Trans. Amer. Math. Soc.

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In this paper we study the tangent space to the Hilbert scheme H i l b d P 3 \mathrm {Hilb}^d \mathbf {P}^3 , motivated by Haiman’s work on H i l b d P 2 \mathrm {Hilb}^d \mathbf {P}^2 and by a long-standing conjecture of Briançon and Iarrobino [J. Algebra 55 (1978), pp. 536–544] on the most singular point in H i l b d P n \mathrm {Hilb}^d \mathbf {P}^n . For points parametrizing monomial subschemes, we consider a decomposition of the tangent space into six distinguished subspaces, and show that a fat point exhibits an extremal behavior in this respect. This decomposition is also used to characterize smooth monomial points on the Hilbert scheme. We prove the first Briançon-Iarrobino conjecture up to a factor of 4 3 \frac {4}{3} , and improve the known asymptotic bound on the dimension of H i l b d P 3 \mathrm {Hilb}^d \mathbf {P}^3 . Furthermore, we construct infinitely many counterexamples to the second Briançon-Iarrobino conjecture, and we also settle a weaker conjecture of Sturmfels in the negative.

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Section: B -Imentioning

confidence: 99%

On the tangent space to the Hilbert scheme of points in 𝐏³

Ramkumar

Sammartano

2022

Trans. Amer. Math. Soc.

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“…In other words, Epdq " K `mr`1 for an ideal K generated by a lexicographic initial segment of monomials of degree r, for some r. It is clear that Epdq " m r when d " `r`2 3 ˘. More generally, the ideal Epdq behaves in many respects in the same way as the powers of m, attaining for every d extremal number of generators, syzygies, socle monomials, and more, see [CS19,V94]. Conjecture 2 stated that Epdq also attains extremal tangent space dimension, for every d. Sturmfels [S00] disproved it by exhibiting counterexamples for d " 8, 16.…”

Section: B -Imentioning

confidence: 99%

On the tangent space to the Hilbert scheme of points in P3

Ramkumar¹,

Sammartano²

2019

Preprint

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A. In this paper we study the tangent space to the Hilbert scheme Hilb d P 3 , motivated by Haiman's work on Hilb d P 2 and by a long-standing conjecture of Briançon and Iarrobino on the most singular point in Hilb d P n . For points parametrizing monomial subschemes, we consider a decomposition of the tangent space into six distinguished subspaces, and show that a fat point exhibits an extremal behavior in this respect. This decomposition is also used to characterize smooth monomial points on the Hilbert scheme. We prove the first Briançon-Iarrobino conjecture up to a factor of 4 3 , and improve the known asymptotic bound on the dimension of Hilb d P 3 . Furthermore, we construct infinitely many counterexamples to the second Briançon-Iarrobino conjecture, and we also settle a weaker conjecture of Sturmfels in the negative. IThe Hilbert scheme of d points in P n , denoted by Hilb d P n , parameterizing closed zerodimensional subschemes of P n of degree d, is a projective moduli space with very rich geometry and a plethora of open questions. It was constructed by Grothendieck [G61] and shown to be connected by Hartshorne [H66]. In the case of P 2 , Fogarty [F68] proved that Hilb d P 2 is nonsingular of dimension 2d, Ellingsurd and Strømme [ES87] computed its homology, and Arcara, Bertram, Coskun, Huizenga [ABCH13] studied its birational geometry in great detail. It also has connections to other areas of mathematics, e.g. to algebraic combinatorics, where it plays a central role in Haiman's proof of the n! Conjecture [H01]. By contrast, the Hilbert scheme is singular for n ě 3 and very little is known about its geometry. The case of Hilb d P 3 is of particular interest, since it lies at the boundary between the smooth cases n ď 2 and the cases n ě 4 which are believed to be wildly pathological, see e.g. [J18]. In fact, Hilb d P 3 is known to be rather special, as it admits a super-potential description -it is the singular locus of a hypersurface on a smooth variety, cf. [BBS13]. For small d, Hilb d P 3 is irreducible, and its general point parametrizes configurations of d points in P 3 ; in particular, the Hilbert scheme has dimension 3d. However, Iarrobino [I72] proved that Hilb d P 3 is reducible for d " 0. In general, the dimension of Hilb d P 3 is unknown. Basic questions about dimension of tangent spaces to Hilb d P 3 are also wide open. Over forty years ago, Briançon and Iarrobino [BI78] established an upper bound for the dimension of Hilb d P n , and stated two conjectures regarding the largest possible dimension of its tangent spaces.Let k be an arbitrary field. For an ideal I, denote by TpIq the tangent space to the corresponding point rIs in the Hilbert scheme. The question of finding the largest possible dimension of a tangent space to Hilb d P n has been raised in many places, including e.g. [BI78, S00, MS05, AIM10]. To answer this question we restrict to an affine open A n " Spec krx 1 , . . . , x n s Ď P n . It is natural to expect that a fat point subscheme V `px 1 , . . . , x n q r ˘Ď A n yields the...

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“…The lexicographic point is always a non-singular point (Theorem 1.6). If the Hilbert scheme has more than one Borel fixed point, the expansive point differs from the lexicographic point [CS19,Proposition 6.6].…”

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“…Caviglia and Sammartano [CS19] introduced another distinguished borel fixed point, called the expansive point. They proved that expansive point is identical to the lexicographic point if and only if the Hilbert scheme has a unique Borel fixed point [CS19,Proposition 6.6]. Since we only need to know it exists and differs from the lexicographic point, we omit its definition.…”

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confidence: 99%

Hilbert schemes with few Borel-fixed points

Ramkumar¹

2019

Preprint

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HILBERT SCHEMES WITH FEW BOREL FIXED POINTS RITVIK RAMKUMAR A. We characterize Hilbert polynomials that give rise to Hilbert schemes with two Borel fixed points and determine when the associated Hilbert schemes or its irreducible components are non-singular. More generally, we show that the Hilbert scheme is reduced and has at most two irreducible components. By describing the singularities in a neighbourhood of the borel fixed points, we prove that the irreducible components are Cohen-Macaulay and normal. We end by giving many examples of Hilbert schemes with three Borel fixed points.

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Syzygies in Hilbert schemes of complete intersections

Cited by 3 publications

References 25 publications

On the tangent space to the Hilbert scheme of points in 𝐏³

On the tangent space to the Hilbert scheme of points in 𝐏³

On the tangent space to the Hilbert scheme of points in P3

Hilbert schemes with few Borel-fixed points

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