Families of Elliptic Curves in P3 and Bridgeland Stability

Gallardo, Patricio; Huerta, César Lozano; Schmidt, Benjamin

doi:10.1307/mmj/1538705132

Cited by 15 publications

(15 citation statements)

References 36 publications

(39 reference statements)

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“…This unique wall leads to a classification of all semistable sheaves, and the description of the moduli spaces follows. The special case uses techniques close to what was done for some Hilbert schemes of curves in [32] and [11].…”

Section: Ingredientsmentioning

confidence: 99%

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Rank two sheaves with maximal third Chern character in three-dimensional projective space

Schmidt¹

2020

RMC

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We give a complete classification of semistable rank two sheaves on three-dimensional projective space with maximal third Chern character. This implies an explicit description of their moduli spaces. As an open subset they contain rank two reflexive sheaves with maximal number of singularities. These spaces are irreducible, and apart from a single special case, they are also smooth. This extends a result by Okonek and Spindler to all missing cases and gives a new proof of their result. The key technical ingredient is variation of stability in the derived category.

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Section: Ingredientsmentioning

confidence: 99%

“…In [29] Okonek and Spindler proved the same bounds for all semistable sheaves of rank two. Moreover, they described the moduli space if either c = 0 and d ≤ −6 or c = −1 and d ≤ − 11 2 . The above theorem fills in all the remaining special cases.…”

Section: Introductionmentioning

confidence: 99%

Rank two sheaves with maximal third Chern character in three-dimensional projective space

Schmidt¹

2020

RMC

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“…Chen [4] proved that the component corresponding to the twisted cubics is the flip of M 0,0 (P 3 , 3) over the Chow variety. Avritzer and Vainsencher [2] proved that the component corresponding to elliptic quartics in Hilb 4t P 3 is smooth and isomorphic to a double blow up of Gr(1, 9); Gallardo, Huerta and Schmidt [13] computed its effective cone. Chen, Coskun and Nollet [5] showed that the component corresponding to a pair of codimension two linear spaces meeting transversely is smooth and isomorphic to a blow of Sym 2 Gr(n − 2, n).…”

Section: Introductionmentioning

confidence: 99%

The Hilbert scheme of a pair of linear spaces

Ramkumar

2021

Math. Z.

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Let $$\mathcal {H}_{a,b}^n$$ H a , b n denote the component of the Hilbert scheme whose general point parameterizes an a-plane union a b-plane meeting transversely in $${\mathbf {P}}^n$$ P n . We show that $$\mathcal {H}_{a,b}^n$$ H a , b n is smooth and isomorphic to successive blow ups of $$\mathbf {Gr}(a,n) \times \mathbf {Gr}(b,n)$$ Gr ( a , n ) × Gr ( b , n ) or $$\text {Sym}^2 \mathbf {Gr}(a,n)$$ Sym 2 Gr ( a , n ) along certain incidence correspondences. We classify the subschemes parameterized by $$\mathcal {H}_{a,b}^n$$ H a , b n and show that this component has a unique Borel fixed point. We also study the birational geometry of this component. In particular, we describe the effective and nef cones of $$\mathcal {H}_{a,b}^n$$ H a , b n and determine when the component is Fano. Moreover, we show that $$\mathcal {H}_{a,b}^n$$ H a , b n is a Mori dream space for all values of a, b, n.

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“…Even when the wall-crossing does not yield the entire MMP, we can still recover crucial information about the birational geometry of the moduli spaces from it, especially when we are able to identify one of them with know projective varieties such as Gieseker moduli spaces. Problems in this sense have been studied for example in [Sch15] and [GLHS16] for the moduli spaces of ideal sheaves of twisted cubics and elliptic quartic curves on P 3 .…”

Section: Introductionmentioning

confidence: 99%