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Algebraic Curves ALGEBRAIC CURVES A 2003 Math Circle session by Renzo INTRODUCTION, MOTIVATION AND GOALS We are going to make a rst few exploratory steps in the fascinating but mysterious world of algebraic geometry. Algebraic geometry is a branch of mathematics that studies geometric problems with the help of algebra (like the more geometrical oriented p eople would say) or algebraic problems with the help of geometry (like the algebraic p eople would say). Whether one opinion is more correct than the other is a minor p oint. What's really co ol is that there is such a close connection b etween these apparently unrelated elds of mathematics. We are going to lo ok in particular in the world of curves, which have the big advantage of b eing \drawable" and hence a little more intuitive and less scary. Algebraic curves have b een studied for a very long time. The ancient Greeks were already pretty handy with them and had gured out a fair amount of things ab out curves that allowed them to solve famous geometrical problems (such as the trisection of the angle or the doubling of the cub e, but this is another story...). Curves are still an extremely interesting ob ject of study nowadays. Even though a LOT of things are known ab out them, there are many questions that are still op en. For example we are still trying to understand the geometry of various mo duli spaces of curves (which you can think of as \spaces of curves with given characteristics". For example, all circles in the plane are identi ed uniquely by the co ordinates of the center and by the radius. Hence we can think that there is a whole three dimensional space of circles: you give me three numb ers, and I will pro duce you a corresp onding circle; you give me a circle, I'll tell you the three corresp onding numb ers). Along the way we'll have a chance to intro duce a \new world", where there are no such things as parallel lines. Again, as bizzarre as this may seem, it's an extremely handy things to do. And it's no funky mo dern invention: the theory of pro jective geometry was initially develop ed by architects and painters during the Renaissance, and heavily used for prosp ectic studies and all sort of applications. Have I convinced you that we are embarking in a worthwile, even though mayb e a little mind quenching adventure? Let's go then!! DEFINITION: An Algebraic Curve consists of the set of p oints in the plane that satisfy a p olynomial equation. 2 C = f(x; y ) 2 R jP (x; y ) = 0g The Degree of the curve is just what we exp ect it to b e: the degree of the p olynomial de ning the curve. EXAMPLES: a line many lines a parabola 2 ax+by+c=0 (what is the equation?) y=x Q1: Given two curves C and C with equations f = 0,f = 0, what is 1 2 1 2 the equation for the curve C [ C ? 1 2 DEFINITION: A curve C is said Irreducible if it do es not break up into the union of \simpler" curves. Q2: How do es this translate in terms of the equation for C ? Q3: Is the equation for an algebraic curve unique? PROBLEM...Our \dictionary" do esn't work so well...We want a go o d corresp ondence b etween algebra (p olynomial equations) and geometry (curves). A dictionary that uses the words banana, bro ccoli and balalaika to mean the exact same thing is not all that go o d, is it? FIX 1:We are using the WRONG numb ers. From now on our numb ers will b e all the complex numb ers and not just the real ones! (Alas, we will have to continue to draw curves in the same way b ecause we yet haven't discovered 4-dimensional chalk!) Notice, though, that now curves will b e \ ob jects of complex dimension 1", which means they really are surfaces. Now mayb e you rememb er that we know surfaces pretty well. We could even work out that all complex algebraic curves are orientable, and hence must lo ok like a sphere or a multiholed doughnut (eventually pinched or punctured), but that's another story.... Q4: What happ ens now to 2 2 x + y = 0 2 2 x + y = 1? Q5: Is the x a complete x? If you p onder a little bit you will nd out that the answer is \almost". In fact we are still de ning the same curve if we multiply the equation by a constant...but that's not such a big deal, and we'll see later on that we'll b e able to x this to o by just slightly mo difying our \universe". For now we have estabilished the following bijective corresp ondence: al g ebr aic cur v es eq uations up to scal ing $ NOW FOR SOME MORE FUN EXAMPLES: Cubics: What are the graphs (of the real trace) of: 2 3 y = x x; 2 3 y = x ; 2 3 2 y = x + x . Pretty Flowers: Match the graphs with their equations: 2 2 3 2 2 (x + y ) 4x y = 0; 2 2 2 (x + y ) + 4xy = 0; 2 2 2 2 3 (x + y ) 3x y y = 0; More pretty owers...: Can you pro duce a sensible guess for the equation of the following gure? Now that we are starting to get a little more familiar with Algebraic Curves, let's start asking ourselvs interesting questions, and let's start easy to then get harder and harder: 1. How many p oints of intersection can 2 lines in the plane have? 2. How many p oints of intersection do a line and a conic have in the plane? 3. How many p oints of intersection do es a line have with a curve C if degree d? 4. How many lines through one p oint? 5. how many lines through two p oints? 6. How many p oints do you need to pin down a unique conic? 7. Will any con guration of the right numb er of p oints pin down a unique conic? 8. What's a quick way to determine the equation of the conic through p 1 1 P = (1; 0), P = (0; 0), P = (0; 1), P = (1; 1), P = ( 2 ; )? 1 2 3 4 5 2 2 9. How many p oints pin down a unique curve of degree d? 2 THE \IDEAL" UNIVERSE : P C Take the usual plane and \add" to it a new (pro jective) line, that we will 2 call line at in nity. Points of this line represent directions of lines in C . point at infinity representing this direction line at infinity Notice that now every pair of lines meet at exactly one p oint!!!! 2 FORMALLY : We add an extra co ordinate, so that p oints in P will b e C identi ed by three complex numb ers (a : b : c): BUT (there's a big but!) we declare two p oints to b e the same p oint if the co ordinates are prop ortional. So, for example (1 : 2 : 3) = (2 : 4 : 6) = (23 : 46 : 69) = (k : 2k : 3k ): Now we can see that 2 2 P = f(a : b : c); c 6= 0g [ f(a : b : 0)g = C [ l : 1 C What happ ens to algebraic geometry now? We could a priori just take 2 p olynomials in three variables (for example P (x; y ; z ) : x + y + 3z + 4 = 0). But obviously that's completely nuts!! Q6: Why? In general I have no hop e to make sense of the notion of the \value" of 2 a p olynomial at a p oint of P . But there are some sp ecial p olynomials such C that at least the zero-set is well de ned. Q7: Can you guess which are they? DEFINITION: A Homogoneous p olynomial is a p olynomial in which all the monomials have the same degree. Q8: Gimme some examples. Q9: Why are homogeneous p olynomials the \go o d ones" for our pur- p oses? DEFINITION: A Pro jective Curve is the zero set of a homogeneous p olynomial. For example: 2 2 2 X + 2XY + Y + 3Z = 0: Q9: Can we go back and forth b etween our previous notion of curves and this current one? The answer is homogenize and dehomogenize !!!! For example: 2 2 2 2 2 x + y 1 = 0 $ X + Y Z = 0; 2 2 y x = 0 $ Y Z X = 0: Q10: Are homogeneization and dehomogeneization always inverse op er- ations to each other? And now for some more problems and interesting stu to discover! 1. Find algebraically the intersection p oints in pro jective space of the following pair of lines: y = x; y = x + 1. 2. Find algebraically the intersection p oints in pro jective space of the following line and hyp erb ola: 0 = x; y x = 1. 2 3. Make sense of the statement: \All conics in P are p otato es". C 2 4. P gives us a real easy way to decide whether a (non pro jective) conic C is a parab ola, ellipse or hyp erb ola. What is it? 5. Decide which of the following conics is a parab ola, ellipse or hyp erb ola! 2 2 x + 2xy + 1 + y + 2y = 0; 2 2 x + 3xy + 2y + 2y + 2x + 6 = 0; 2 2 x + xy + y + 4y + 3x 12 = 0.
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