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Moduli spaces of quasitrivial sheaves on the three dimensional projective space

Abstract : Let M(r,c_1,c_3,c_3) denote the Gieseker--Maruyama moduli space of semistable rank r sheaves on P^3 with the first, second and third Chern classes equal to c_1, c_2 and c_3, respectively. Maruyama proved that the space M(r,c_1,c_3,c_3) is a projective scheme. However, the geometry of such a scheme remains largely unknown, despite the efforts of many authors in the past four decades, and questions about connectedness, irreducibility, the number of irreducible components, and so on, remain open.When r=1 and c_1=0 (which can always be achieved after twisting by an appropriate line bundle), one gets that M(1,0,c_2,c_3) is isomorphic to the Hilbert scheme Hilb^{d,g}(P^3) of 1-dimensional schemes of degree d=-c_2 and genus g=c_3-2c_2, which is known to always be connected. Not much is known in general when r>1, though:- M(2,c_1,c_2,c_3) is irreducible for c_3=c_2^2-c_2+2 when c_1=0, or c_3=c_2^2 when c_1=-1;- M(2,0,2,c_3) has 2 irreducible components when c_3=2 and it has 3 irreducible components when c_3=0.- M(2,-1,2,c_3) has 2 irreducible components when c_3=2 and it has 4 irreducible components when c_3=0.Moreover, the moduli spaces in items (2) and (3) are connected. For higher values of c_2, one can check that the number of irreducible components of M(2,c_1,c_2,0) grows with c_2; it is not known whether M(2,c_1,c_2,c_3) is always connected.The goal of this work is to explore a somewhat exotic case, namelyM(r,0,0,-2n)=:N(r,n),whose points correspond to quasitrivial rank r sheaves, that is, semistable rank r sheaves E on P^3 such that E**=O^r and dim(E**/E)=0; this nomenclature is borrowed from Artamkin. The motivation comes from its close relationship, described in the body of the paper, between N(r,n) and the Hilbert and Quot schemes of points in P^3. Moreover, even though the main focus of this work is the moduli space of semistable quasitrivial sheaves, we also provide some results regarding mu-semistable quasitrivial sheaves.First, we study mu-semistable sheaves E on P^d with rk(E)>0 and c_1(E)=c_2(E)=0, and show that they are always extensions of ideal sheaves of subschemes of P^d of codimension at least 3. In addition, we prove that the moduli space of such sheaves is a GIT quotient of a Quot scheme, where the Hilbert polynomial have degree less than or equal to d-3.We then focus on the case d=3, for which we can get more concrete and precise results. Here is the main result of this work.- N(r,n) is empty whenever r>n or n<0.- N(n,n) is isomorphic to Sym^n(P^3).- If r
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Submitted on : Wednesday, November 23, 2022 - 8:50:09 AM
Last modification on : Friday, November 25, 2022 - 10:12:06 AM

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Douglas Guimaraes. Moduli spaces of quasitrivial sheaves on the three dimensional projective space. General Mathematics [math.GM]. Université Bourgogne Franche-Comté; Universidade estadual de Campinas (Brésil), 2022. English. ⟨NNT : 2022UBFCK003⟩. ⟨tel-03866936⟩

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