TITLE: Algebraic geometry examples
QUESTION [77 upvotes]: What are some surprising or memorable examples in algebraic geometry, suitable for a course I'll be teaching on chapters 1-2 of Hartshorne (varieties, introductory schemes)? 
I'd prefer examples that are unusual or nonstandard, as I already know many of the standard ones (27 lines on a cubic surface, etc). 
To illustrate the sorts of things I am looking for, here are some examples that would have been useful answers if I had not already thought of them: 
The  proof of the Kakeya conjecture for finite fields.
Free sheaves need not be projective
The Hilbert scheme of m points on an n dimensional variety can have dimension larger than mn.
The scheme of nilpotent matrices is not reduced. 
The 1-line proof of Pascal's theorem from Bezout's theorem.
Spec Z[x]
Resolution of the Whitney umbrella
The related threads
What should be learned in a first serious schemes course? and Interesting results in algebraic geometry accessible to 3rd year undergraduates also have some good examples, such as Grassmannians.
Added later: Thanks for all the examples; I'm embarrassed to admit that some of them are counterexamples to statements I would have guessed were true. Over the next few weeks I will  gradually add answers below (with credit to those who suggested them) to my  draft course notes (These notes still need a lot of corrections/expansion/rewriting; they should have reached a more stable state by Dec 2010)

REPLY [4 votes]: Given in an order from less challenging to do in such a course to more challenging.

Conics (to "connect" with high-school algebraic geometry)
All smooth conics in $\mathbb{P}^2_{k}$ that have a rational point are isomorphic to $\mathbb{P}^1$. In particular, ellipse, hyperbola and parabola are the same; one may also learn how to distinguish them in the affine plane over $\mathbb{R}$.
Steiner's construction of a conic through 5 points, no three of which are collinear is an example of how (synthetic) projective geometry can dispense with algebra! If you are adventurous, you can also give Steiner's construction of the rational normal curve.

Flatness (example learnt from S. Ramanan)
The morphism from $V((x)\cap(x-t))$ in $\mathbb{A}^2$ to $\mathbb{A}^1$ given by $(x,t)\mapsto t$ is flat.
However, the morphism from $V((x,y)\cap(x-t,y-s))$ in $\mathbb{A}^4$ to $\mathbb{A}^2$ given by $(x,y,t,s)\mapsto (s,t)$ is not flat.
This is because the "degenerate doubled point" in the plane must keep track of the "direction" in which it was doubled.
The "real" flat family of "pairs of points" in the plane is parametrized by the blow-up of $\mathbb{A}^2$ at a point.
The insight that "flatness" is an essential condition for extending the notion of families to singular situations was a "game-changer" for algebraic geometry. Flatness is a definition that seems to require algebra and yet is important for geometry.

Tangent bundle.
The tangent bundle of a smooth variety $X$ over $k$ is the scheme of $k[\epsilon]$ points of $X$. (This is an extension of the above example, in a way.)
This example clearly shows how infinitesimals have a meaning in algebraic geometry.

Hirzebruch surfaces (learnt from Beauville's book on surfaces).
Given the rational normal curve $C_k$ in $\mathbb{P}^k$ and the rational normal curve $C_m$ in $\mathbb{P}^m$. Choose an isomorphism $g$ between $C_k$ and $C_m$, and choose disjoint linear embeddings of $\mathbb{P}^k$ and $\mathbb{P}^m$ in $\mathbb{P}^n$ (so $n\geq k+m+1$). Let $J$ be the union of the lines joining $g$-pairs of points in $C_k$ and $C_m$.
Then $J$ is the Hirzebruch surface $F_{|m-k|}$. Moreover, if $m<k$ then $C_m$ is the curve on $J$ with negative self-intersection.
Bonus points for understanding "elementary transformations" $F_{a}\to F_{a\pm 1}$ in terms of projection of this surface from a point on $C_k$ or $C_m$!

Quartic curve.
Redo the analysis of a quartic equation from Chapter 12 (section 95+, page 351+) of Weber's book on "Algebra" in modern language. He demonstrates the existence of 28 bitangents. It is completely "high-school" algebra and geometry argument which does not involve and Abelian varieties and theta characteristics! Essentially he re-write the equation in the form $\sqrt{a_1a_2}+\sqrt{b_1b_2}+\sqrt{c_1c_2}$ where $a_i, b_i, c_i$ are linear forms.
In my humble opinion, this is more challenging and interesting than 27 lines on a cubic surface. In fact, the latter follows from the above analysis by doubling the plane along the quartic curve.