A blowup formula for virtual enumerative invariants on projective surfaces

The Blowup Formula for the Instanton Part of Vafa-Witten Invariants on Projective Surfaces

“…In [NY11c], Nakajima and Yoshioka introduced an algorithm to express certain tautological integrals on M ( c) in terms of similar integrals on M (c + n[pt]) for various n, which applies in Situation B. The results necessary to apply the algorithm in Situation A were established in [KT21]. In this section we will review the Nakajima-Yoshioka blow-up algorithm in both situations simultaneously and draw some conclusions.…”

“…Situation A -Gieseker stable sheaves [Moc09,KT21]. In this situation, X is a smooth projective (connected) surface with a fixed polarisation H and p : X → X is the blow-up of X at a chosen reduced point pt ∈ X with exceptional divisor C. Moreover, L X is a fixed line bundle on X with c 1 := c 1 (L X ), such that the intersection number (L X , H) is coprime to r.…”

“…In the setting of Theorem 1.4, using appropriately defined generating series, we have Outline of the Article. In order to prove Theorem 1.4, we use the blow-up algorithm of Nakajima-Yoshioka [NY11c] (extended to Situation A by [KT21]) to obtain a weak universal blowup formula for invariants of the form being studied.…”

“…• Section 2: We recall the blow-up algorithm of Nakajima-Yoshioka [NY11c] and show how it is extended to both situations by [KT21]. • Section 3: We state and prove a weak universal blowup formula for invariants defined by classes which are multiplicative in the tangent bundle.…”