Applications to $${\mathbb {A}}^1$$-enumerative geometry of the $${\mathbb {A}}^1$$-degree

Abstract

These are lecture notes from the conference Arithmetic Topology at the Pacific Institute of Mathematical Sciences on applications of Morel's A 1 -degree to questions in enumerative geometry. Additionally, we give a new dynamic interpretation of the A 1 -Milnor number inspired by the first named author's enrichment of dynamic intersection numbers.

An enriched degree of the Wronski

“…Recent results include an enriched Bézout's theorem [McK21], an enriched count of lines on a quintic threefold [Pau20], and a count of conics meeting eight lines [Dar+21]. For further reading on this field we refer the reader to the survey papers [Bra21;PW21].…”

“…We can also ask about how we might wield Nisnevich coordinates to compute a global A 1 -degree of a morphism between suitably nice smooth n-schemes. This notion is based off forthcoming work of Kass, Levine, Solomon and Wickelgren [Kas+22], and was discussed briefly in the expository paper [PW21,§8]. See also [MS20, p. 2.53] for the degree of an endomorphism of projective space, and [Mor12] for the degree of an endomorphism of P n /P n−1 .…”

“…Welschinger [Wel05] remarked that (−1) #I is a revealing invariant to consider for planar curves, where I is the set of isolated points. Kass-Levine-Solomon-Wickelgren have extended this to define an arithmetic Welschinger invariant valued in GW(k) [Kas+22] (see also [Lev18] and [PW21]). We may compute that Welschinger's original invariant agrees with the local index of σ following the formula in Theorem 4.3.…”