Affine pavings for moduli spaces of pure sheaves on $$\mathbb {P}^2$$ P 2 with degree $$\le 5$$ ≤ 5

Yao, Yuan

doi:10.1007/s10711-014-9995-x

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“…Remark 1.6. Theorem 1.3 and Theorem 1.4 give the motive decompositions of M(d, r) for d = 4, 5, r coprime to d. But according to the result in [11] that these moduli spaces admit affine pavings, we also get cell decompositions of them.…”

Section: Introductionmentioning

confidence: 96%

Moduli Spaces of Semistable Sheaves of Dimension $1$ on $\mathbb{P}^2$

Yao

2014

Pure and Applied Mathematics Quarterly

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Let M(d, χ) be the moduli space of semistable sheaves of rank 0, Euler characteristic χ and first Chern class dH (d > 0), with H the hyperplane class in P 2 . We give a description of M(d, χ), viewing each sheaf as a class of matrices with entries in i≥0 H 0 (O P 2 (i)). We show that there is a big open subset of M(d, 1) isomorphic to a projective bundle over an open subset of a Hilbert scheme of points on P 2 . Finally we compute the classes of M(4, 1), M(5, 1) and M(5, 2) in the Grothendieck ring of varieties, especially we conclude that M(5, 1) and M(5, 2) are of the same class.Theorem 1.3 (Theorem 5.2). [M(4, 1)] = 17 i=0 b 2i L i and b 0 = b 34 = 1, b 2 = b 32 = 2, b 4 = b 30 = 6, b 6 = b 28 = 10, b 8 = b 26 = 14, b 10 = b 24 = 15, b 12 = b 14 = b 16 = b 18 = b 20 = b 22 = 16.In particular the Euler number e (M(4, 1)) of the moduli space is 192. Theorem 1.4 (Theorem 6.1). [M(5, 1)] = [M(5, 2)] = 26 i=0 b 2i L i and b 0 = b 52 = 1, b 2 = b 50 = 2, b 4 = b 48 = 6, b 6 = b 46 = 13, b 8 = b 44 = 26, b 10 = b 42 = 45, b 12 = b 40 = 68, b 14 = b 38 = 87, b 16 = b 36 = 100, b 18 = b 34 = 107, b 20 = b 32 = 111, b 22 = b 30 = 112, b 24 = b 26 = b 28 = 113.In particular the Euler number of both moduli spaces is 1695.We then have µ(E) + 1 = µ(F ) for E, F in the sequence (3.1).Definition 3.1. We say a pair (E, f ) is (semi)stable if for any subsheaf E ′ E and E ′ a direct sum of line bundles such that f −1 (E ′ ) ≃ E ′ ⊗ O P 2 (−1), we have µ(E ′ )(≤) < µ(E).Lemma 3.2. θ induces a bijection from semistable pairs to semistable sheaves.

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Section: Introductionmentioning

confidence: 96%