1. Normal affine toric varieties and polyhedral cones (14.04.15).

Aim of the talk: Introduce normal affine toric varieties and explain the connection with cones.

Details: The content and notation should cover [CLS11, 1.2]. Start with [CLS11, Def. 1.1.3], but then stick to the normal case. Explain the different terms (e.g. open subset, multiplication, etc.) in terms of graded commutative algebra.

First introduce lattices, their dual and the associated vector space. Introduce the notion of (pointed) rational convex polyhedral cones, define their dual, faces and interior (see p. 6-7 in I, §1). Do simple examples as $N=\mathbb{Z}^2\cong M$, $N_{\mathbb{Q}}=\mathbb{Q}^2$ and take for $\sigma$ the positive orthant. Take the associated semigroup and the semigroup algebra. Show that it is a subalgebra of $k[M]$ via $\sigma^\vee\subseteq M$. Explain finite generation of $\sigma^\vee\subseteq M$ (Hilbert basis, toric ideal, [CLS11, Prop 1.2.17]). Try to emphasize the commutative algebra side. Also do the orbit-face correspondence [KKMSD73, Theorem 2], but in the notation of [CLS11, Thm. 3.2.6]. Decompose $\mathbb{C}^2$ and a CQS.

Do the examples [Cox, Ex. 1.1- 1.4] (and more examples if time permits).

Anna-Lena