I will be giving a talk to a (primarily) undergraduate audience on certain relatively concrete computations with toric varieties and their blowups. The talk is short, about 20 mins. As I result I need to introduce toric varieties (smooth, projective) and I would like to have my audience understand that the combinatorics of a fan/polytope can be used to understand the geometry of the toric variety.


When I've given similar talks to similar audiences in the past, I have been very hand-wavey just saying "there's a copy of $(\mathbb{C}^\*)^n$ inside my variety and as a consequence we have this magical polytope encoding all this information." 


The Question: How would you motivate and introduce toric varieties quickly? Is there some classical hands-on example that you find particularly illustrative?



Clean introduction to toric varieties for an undergraduate audience
David Cox has some nice expositions on toric varieties on his web page [here](http://www3.amherst.edu/~dacox/). Cox is also one of the authors of the book "Toric Varieties", which is a very readable, yet comprehensive introduction to toric varieties. The first chapter here should provide you with enough motivation and examples for your talk. Then there is also chapter 1 in Fulton's book, which is the classic reference on the subject.


As for the motivational examples, you should look for examples that show the real power of toric varieties: *That abstract algebro-geometric constructions can uaually be viewed very concretely by working with the defining combinatorial data (e.g. the fan)*. Some of these examples might do the trick:


1) The quadric surface $\{xy-zw=0\}$ in $\mathbb P^3$ and its affine cone in $\mathbb A^4$


2) The singular quadric $y^2=zw$ in $\mathbb A^3$. 


3) Hirzebruch surfaces 


4) Toric blow-ups and subdivisons of the fan


In the basic examples 1)-3), it is straightforward to write out the action of the torus, and see directly how monomials in the coordinate ring relates to the lattice points in the dual cones. Also, in the projective examples you can see how gluing the affine toric varieties works in terms of the fan data. 


These examples demonstrate typical features of toric varieties, for example that their ideals are generated by binomials and their Chow ring is generated by the torus invariant subvarieties. 


I like the Hirzebruch surface example because you can somehow 'see' the $\mathbb P^1$-bundle structure in the defining polytope and it is intuiticely clear that any toric surface is a blow-up of either $\mathbb P^2$ or a Hirzebruch surface. Moreover, I think it's pretty cool that you can view birational morphisms of toric varieties (e.g., resolution of singularities) as subdivisions of the defining fans. The example in this [MO thread](https://mathoverflow.net/questions/64181/blowing-curves-down/64203#64203) illustrates this. Another interesting example is the affine cone $Z(xy-zw=0)\subset \mathbb A^4$ which gives a nice combinatorial interpretation of the Atiyah flop.


