I: Definitions and Examples

Machinery used to study varieties:

• Various cohomology theories
• Resolutions of singularities
• Intersection theory and cycles
• Riemann-Roch theorems
• Vanishing theorems
• Linear systems (via line bundles and projective embeddings)

Varieties that arise as examples

• Grassmannians
• Flag varieties
• Veronese embeddings
• Scrolls
• Quadrics
• Cubic surfaces
• Toric varieties (of course)
• Symmetric varieties and their compactifications

Misc notes:

• Toric varieties are always rational

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• Toric varieties: normal varieties \(X\) with \(T\hookrightarrow X\) contained as a dense open subset where the torus action \(T\times T\to T\) extends to \(T\times X\to X\).
• Any product of copies of \({\mathbf{A}}^n, {\mathbf{P}}^m\) are toric.
• \(S_\sigma\) is a finitely-generated semigroup, so \({\mathbf{C}}[S_\sigma] \in \mathsf{Alg}{{\mathbf{C}}}^{\mathrm{fg}}\) corresponds to an affine variety \(U_\sigma \coloneqq\operatorname{Spec}{\mathbf{C}}[S_\sigma]\).
• If \(\tau \leq \sigma\) is a face then there is a map of affine varieties \(U_\tau \to U_\sigma\) where \(U_\tau = D(u_\tau)\) is a principal open subset given by the function \(u_\tau\) picked such that \(\tau = \sigma \cap u_\tau^\perp\), so \(u_\tau\) corresponds to the orthogonal normal vector for the wall \(\tau\).
• These glue to a variety \(X(\Delta)\).
• Smaller cones correspond to smaller open subsets.
• The geometry in \(N\) is nicer than that in \(M\), usually.
• Rays \(\rho\) correspond to curves \(D_\rho\).

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• Convex polyhedral cones: generated by vectors \(\sigma = {\mathbb{R}}_{\geq 0}\left\langle{v_1,\cdots, v_n}\right\rangle\). Can take minimal vectors along these rays, say \(\rho_i\).

• \(\dim \sigma \coloneqq\dim_{\mathbb{R}}{\mathbb{R}}\sigma \coloneqq\dim_{\mathbb{R}}(-\sigma + \sigma)\)
• \((\sigma {}^{ \vee }) {}^{ \vee }= \sigma\), which follows from a general theorem: for \(\sigma\) a convex polyhedral cone and \(v\not\in \sigma\), there is some support vector \(u_v\in \sigma {}^{ \vee }\) such that \({\left\langle {u},~{v} \right\rangle} < 0\). I.e. \(v\) is on the negative side of some hyperplane defined in \(\sigma {}^{ \vee }\).
• Faces are again convex polyhedral cones, faces are closed under intersections and taking further faces.
• If \(\sigma\) spans \(V\) and \(\tau\) is a facet, there is a unique \(u_\tau\in \sigma {}^{ \vee }\) such that \(\tau = \sigma \cap u_\tau^\perp\); this defines an equation for the hyperplane \(H_\tau\) spanned by \(\tau\).
• If \(\sigma\) spans \(V\) and \(\sigma\neq V\), then \(\sigma = \cap_{\tau\in \Delta} H_\tau^+\), the intersection of positive half-spaces.
  • An alternative presentation: picking \(u_1,\cdots, u_t\) generators of \(\sigma {}^{ \vee }\), one has \(\sigma = \left\{{v\in N {~\mathrel{\Big\vert}~}{\left\langle {u_1},~{v} \right\rangle} \geq 0, \cdots, {\left\langle {u_t},~{v} \right\rangle}\geq 0}\right\}\).
• If \(\tau \leq \sigma\) then \(\sigma {}^{ \vee }\cap\tau {}^{ \vee }\leq \sigma {}^{ \vee }\) and \(\dim \tau = \operatorname{codim}(\sigma {}^{ \vee }\cap\tau {}^{ \vee })\), so the faces of \(\sigma, \sigma {}^{ \vee }\) biject contravariantly.
• If \(\tau = \sigma \cap u_\tau^\perp\) then \(S_\tau = S_\sigma + {\mathbb{N}}\left\langle{-u_\tau}\right\rangle\).