IRREDUCIBILITY AND SMOOTHNESS OF THE MODULI SPACE OF MATHEMATICAL 5-INSTANTONS OVER ℙ₃

Abstract

We prove that the space of mathematical instantons with second Chern class 5 over P 3 is smooth and irreducible. Unified and simple proofs for the same statements in case of second Chern class ≤ 4 are contained.

Singular loci of instanton sheaves on projective space

“…The previous Proposition allows us to provide a neat description of the moduli space of indecomposable rank 3 instanton sheaves of charge 1, which we will denote here by I tf (3,1). If E is such an object, let ℘ E be the corresponding hyperplane in P 3 , obtained via the sequence (34); this yields a map…”

“…The simplest case of such objects are rank 2 instanton bundles on CP 3 , and there is a vast literature about them. One outstanding problem that resisted solution for a couple of decades regards the irreducibility and smoothness of the moduli space I(c) of rank 2 instanton bundles on CP 3 of charge c. It was known since 2003 that I(c) is smooth and irreducible for c ≤ 5, see [1] and the references therein. Recently, Tikhomirov has proved in [14,15] irreducibility for arbitrary c; while the second named author and Verbitsky established smoothness for every c, see [8].…”

Monads and instanton bundles on smooth hyperquadrics

“…The existence of the moduli space M I P 2n+1 (k) of instantons bundle on P 2n+1 with quantum number k was established by Okonek and Spindler in [16, Theorem 2.6]. Determining the irreducibility and smoothness of M I P 2n+1 (k) is a long standing question far from being solved, see [5] and [11] for a recent survey on the topic.…”

Moduli of autodual instanton bundles

“…In dimension 3, the nonsingularity and irreducibility of the moduli space of rank 2 instantons bundles on P 3 remained open for many years, see for instance [5,Introduction] and the references therein. More recently, Tikhomirov proved in [18,19] that the moduli space of rank 2 instanton bundles on P 3 with arbitrary second Chern class is irreducible; Markushevich and Tikhomirov proved that it is rational; Jardim and Verbitsky proved in [12] that such moduli spaces are always nonsingular.…”