Where we are on the ladder
You arrive at the algebraic-geometry tower already fluent in several others. You know what a topological space is and what open sets, compactness, and connectedness do for you; you have done real and complex analysis; and you carry a working command of commutative algebra — rings, ideals, fields, and quotients. This rung adds a genuinely new instinct: stop thinking of a curve as the set of points a moving particle traces, and start thinking of it as the solution set of polynomial equations. That single change of viewpoint is what separates differential geometry, where smooth bump functions are everywhere, from algebraic geometry, where the only functions allowed are polynomials and there are dramatically fewer of them.
An honesty note before we start. These are graduate topics, and we assume comfort with linear algebra, multivariable calculus, point-set topology, and abstract algebra — especially ideals in polynomial rings. We will work over an algebraically closed field k (think of the complex numbers C, where every nonconstant polynomial has a root); much of the cleanest theory simply fails over the reals, and we will be explicit each time that closedness is doing the work. This first guide builds the geometric stage — what the objects are and what topology they wear. The bridge that turns geometry into algebra, Hilbert's Nullstellensatz, is the entire subject of guide 2; here we set it up and quote only the bare minimum.
Affine varieties: shapes cut out by polynomials
Start in affine n-space, written A^n, which is just k^n thought of as a stage for geometry rather than as a vector space (we forget the privileged origin). An affine variety is the common zero set of a collection of polynomials: given polynomials f_1, ..., f_r in k[x_1, ..., x_n], the variety V(f_1, ..., f_r) is the set of points p in A^n where every f_i(p) = 0. A single equation y - x^2 = 0 gives a parabola; the pair x^2 + y^2 + z^2 - 1 = 0 inside A^3 gives a sphere; the empty collection gives all of A^n. The whole point of an affine variety is that it is defined by equalities of polynomials and nothing else — no inequalities, no limits, no smoothness assumptions.
A crucial bookkeeping fact tames the apparent freedom of choosing polynomials. The zero set of a family {f_i} is unchanged if we throw in any polynomial of the form g_1 f_1 + ... + g_r f_r, because such a combination automatically vanishes wherever all the f_i do. The set of all these combinations is exactly the ideal generated by the f_i, so the variety only ever depends on the ideal I = (f_1, ..., f_r), not on the particular generators. Hilbert's basis theorem then guarantees every ideal in k[x_1, ..., x_n] is finitely generated, so 'a collection of polynomials' and 'a finite list of polynomials' carve out the same shapes — there are no varieties needing infinitely many equations.
The Zariski topology: closed sets first, and very coarse
We now make A^n a topological space in the only way the polynomials suggest: declare the closed sets to be exactly the affine varieties. One checks the axioms quickly — the whole space A^n = V(0) and the empty set = V(1) are closed; an arbitrary intersection of varieties is V of the sum of their ideals, hence a variety; and a finite union V(I) ∪ V(J) equals V(IJ), again a variety. (Only finite unions survive — that is the algebra of products talking.) This is the Zariski topology, and a Zariski-closed set is precisely a variety. The open sets are the complements: the things you get by removing a variety from A^n.
This topology will feel deeply wrong at first, and you should let it. On the affine line A^1, the proper varieties are the finite sets of points (the zeros of a single one-variable polynomial), so the closed sets are: the whole line, and every finite set. The nonempty open sets are therefore the cofinite sets — the line minus finitely many points. Any two nonempty open sets must intersect, since two cofinite sets cannot be disjoint, so A^1 is not Hausdorff: you cannot separate two points by disjoint opens. The Zariski topology is far coarser than the metric topology you know from analysis; it remembers algebra, not distance.
Irreducibility: the right notion of a single piece
In the metric world a connected set is 'one piece,' but the Zariski topology is too coarse for connectedness to be the useful notion — A^1 is Zariski-connected yet we still want to call the variety V(xy) = (the two coordinate axes) 'two pieces.' The right notion is irreducibility: a nonempty variety is irreducible if it cannot be written as a union of two strictly smaller closed sets. The union of the two axes V(xy) = V(x) ∪ V(y) is reducible; a line, a parabola, or all of A^n is irreducible. Algebraically, a variety is irreducible exactly when its defining ideal is prime, which is the cleanest dictionary entry we will meet before the Nullstellensatz proper.
Every variety then breaks uniquely into irreducible pieces. The Noetherian property of k[x_1, ..., x_n] (no infinite strictly-descending chain of varieties) forces any variety to be a finite union of maximal irreducible closed subsets, and these are its irreducible components — the algebraic analogue of connected components, but finer. The two axes have two components; a nodal cubic like y^2 = x^2(x+1) is irreducible (one component) even though it crosses itself at the origin. Decomposing into irreducible components is almost always the first move in studying any variety, because the deepest theory (dimension, function fields, smoothness) is cleanest on an irreducible variety.
Projective varieties: closing up at infinity
Affine space has an annoying defect: things run off to infinity and never come back. Two parallel lines never meet; a line and a hyperbola can miss each other; a conic can have 'fewer' intersections than its degree predicts. The cure is to compactify by adding points at infinity, giving projective n-space P^n. A point of P^n is a line through the origin in k^(n+1), written in homogeneous coordinates [a_0 : a_1 : ... : a_n] where not all a_i are zero and where rescaling all coordinates by the same nonzero lambda gives the same point. The original affine A^n sits inside as the points with a_0 ≠ 0; everything with a_0 = 0 is the 'hyperplane at infinity' that was missing.
What can 'a polynomial vanishing' even mean on P^n, where a point has no single coordinate value? Only for homogeneous polynomials does it make sense: if F is homogeneous of degree d then F(lambda·a) = lambda^d · F(a), so 'F(a) = 0' does not depend on which representative of the point we picked. A projective variety is the common zero set in P^n of a collection of homogeneous polynomials. The dictionary to the affine world is concrete: to pass from a projective variety to its affine part, set a_0 = 1 (dehomogenize); to go back, multiply each monomial by the right power of a_0 so every term has the same degree (homogenize). Each affine chart {a_i ≠ 0} is a copy of A^n, and P^n is glued from n+1 such charts.
Affine parabola in A^2 : y - x^2 = 0 Homogenize (x=X/Z, y=Y/Z) : Y Z - X^2 = 0 in P^2 [X : Y : Z] Points at infinity (Z = 0) : X^2 = 0 => [0 : 1 : 0] So the affine parabola gains exactly ONE point at infinity, where both its 'ends' run together — the projective curve is a circle (topologically).
Why projective is the natural home, and what comes next
Projective space carries its own Zariski topology (closed sets = projective varieties), and over C it is compact in the ordinary topology too — that compactness is exactly what 'closing up at infinity' buys. The payoff is that intersection theory finally behaves. In P^2 over an algebraically closed field, two projective plane curves of degrees m and n meet in exactly m·n points, counted with the right multiplicities — this is Bézout's theorem, the climax of guide 5, and it is simply false in the affine plane where intersection points can escape to infinity. Parallel lines now do meet, at a shared point at infinity; the missing intersections of a conic reappear. Projective geometry is where the count is honest.
Two honest cautions before we move on. First, a variety is not the same as its equations: many different ideals can cut out the same point set, and sorting out which ideal a variety 'remembers' is precisely the question the Nullstellensatz answers next guide — the short version is that a variety only sees an ideal up to its radical, so x = 0 and x^2 = 0 cut out the same affine variety even though the second 'should' record a double point. Holding that distinction loosely now is fine; pinning it down is guide 2. Second, do not over-trust the slogan 'projective = compact + good intersection theory': compactness over C is genuine, but Bézout still needs the field algebraically closed, the curves with no common component, and multiplicities counted correctly — drop any of those and the clean count breaks.
A clean way to hold this guide in memory: algebraic geometry studies shapes that are zero sets of polynomials, topologized by the coarse Zariski topology where closed means 'cut out by equations,' decomposed into irreducible components, and best housed in projective space where nothing escapes to infinity. From this stage the rest of the rung unfolds: guide 2 builds the exact ideal-variety dictionary, guide 3 attaches the coordinate ring and function field that make a variety into something you can do calculus-free analysis on, guide 4 measures dimension and tells smooth points from singular ones, and guide 5 cashes everything out in Bézout's count. Equations first, geometry as their shadow — that is the whole reorientation.