Commutative algebraic monoid structures on affine spaces

Abstract

We study commutative associative polynomial operations A n ×A n → A n with unit on the affine space A n over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.

Additive actions on complete toric surfaces

“…There is a number of results on additive actions on flag varieties [1,[13][14][15], singular del Pezzo surfaces [12], Hirzebruch surfaces [17] and weighted projective planes [2].…”

Classification of noncommutative monoid structures on normal affine surfaces

“…which is the same as the comultiplication in (1). Since the cone σ ∨ has ray generators (a, b) and (0, 1) with a > 0 and b ≥ 0, this is a comultiplication, which turns Z σ into a monoid, which is isomorphic to X a,b n .…”

“…In the basis from Lemma 3.13 we have δ e = ∂ χ (1,0) . This basis plays a crucial role in the proof of the next theorem.…”

“…Suppose now that n < 0. This time write x for χ (1,0) and y for χ (0, −1) . Using the same argument, we obtain the comultiplication in (1) and the corresponding monoid structure is isomorphic to Y a,b+na Proof.…”

“…However, there were no attempts to classify general affine algebraic monoids even in low dimensions until recently. In [1], the authors studied commutative monoid structures on affine surfaces. They gave explicit formulas for possible multiplications on A 1 , A 2 , and A 3 .…”