1 16.01. Lecture 1. Complex algebraic curves in C2 and real algebraic curves Let P (x; y) 2 C[x; y] be a polynomial in 2 variables with complex coefficients. P has no repeated factors if it cannot be written as P (x; y) = Q(x; y)2R(x; y) for non-constant polynomial Q. Definition 1.1. Let P 2 C[x; y] be a non-constant polynomial with no repeated factors. The complex algebraic curve defined by P is 2 C = f(x; y) 2 C j P (x; y) = 0g: We want to assume that curves are given by polynomials with no repeated factors because of the following Theorem 1.2 (Hilbert's Nullstellensatz). If P (x; y) and Q(x; y) are polynomials in C[x; y] then 2 2 f(x; y) 2 C j P (x; y) = 0g = f(x; y) 2 C j Q(x; y) = 0g if and only if there exist m; n 2 N such that P divides Qm and Q divides P n. In other words, if and only if P and Q have the same irreducible factors possibly occurring with different multiplicities. It follows from Theorem 1.2 that two polynomials P , Q with no repeated factors define the same complex algebraic curve C if and only if there exists λ 2 C∗ such that P = λQ. In general, one can define complex algebraic curves as equivalence classes of non- constant polynomials P 2 C[x; y] where P ∼ Q if and only if P and Q have the same irreducible factors. A polynomial with repeated factors defines a curve with multiplicities attached, for P = x2y the line fy = 0g has multiplicity one while fx = 0g comes into C = fP (x; y) = 0g with multiplicity two. Definition 1.3. The degree of a curve C = fP (x; y) = 0g is the degree of the polynomial P r s P (x; y) = cr;sx y , i.e. d = maxfr + s j cr;s 6= 0g: Definition 1.4. A point (a; b) 2 C is a singular point (or singularity) of C, if @P @P (a; b) = 0; (a; b) = 0: @x @y The set of singular points is denoted by Sing(C). C is called non-singular if Sing(C) = ;. Definition 1.5. A curve defined by αx + βy + γ = 0; where α; β; γ 2 C and (α; β) 6= (0; 0), is a line. 1 We will be mostly interested in curves in projective plane. They are given by homogeneous polynomials Definition 1.6. A non-zero polynomial P 2 C[x1; : : : ; xn] is homogeneous of degree d if d P (λx1; : : : ; λxn) = λ P (x1; : : : ; xn); for all λ 2 C. Equivalently P has the form X r1 rn P (x1; : : : ; xn) = ar1;:::;rn x1 : : : xn : r1+:::+rn=d Lemma 1.7. If P 2 C[x; y] is homogeneous of degree d then it factors as a product of linear polynomials d Y P (x; y) = (αix + βiy); i=1 for some αi; βi 2 C. Proof. We have d d X X X x P (x; y) = a xrys = a xryd−r = yd a ( )r: r;s r;d−r r;d−r y r+s=d r=0 r=0 Pd r Polynomial Pe(t) := r=0 ar;d−rt 2 C[t] is a complex polynomial in one variable, hence it can be factorised as d e X r Y Pe(t) = ar;d−rt = ae (t − γi); r=0 i=1 where e = maxfr j ar;d−r 6= 0g is the degree of Pe. Then e e d x d Y x d−e Y P (x; y) = y Pe( ) = y ae (( ) − γi) = aey (x − γiy): y y i=1 i=1 Since P (x; y) is a polynomial it has a finite Taylor expansion X @i+jP (x − a)i(y − b)j P (x; y) = (a; b) @xi@yj i!j! i;j≥0 at any point (a; b) 2 C2. Definition 1.8. The multiplicity of curve C = fP (x; y) = 0g at point (a; b) 2 C is the @mP smallest m 2 N such that @xi@yj (a; b) 6= 0. The polynomial X @mP (x − a)i(y − b)j (a; b) (1) @xi@yj i!j! i+j=m 2 is then homogeneous of degree m and so by Lemma 1.7 can be written as a product of polynomials of the form α(x − a) + β(y − b) where (α; β) 2 C2 n f0g. The lines defined by these linear polynomials are tangent lines to C at (a; b). The point (a; b) is non-singular if and only if its multiplicity is 1; in this case C has just one tangent line @P @P (a; b)(x − a) + (a; b)(y − b) = 0: @x @y A point (a; b) 2 C is a double point (respectively triple point, etc.) if the multiplicity of C at (a; b) is two (respectively three, etc.). A singular point is ordinary if polynomial (1) has no repeated factors, i.e. if C has m distinct tangent lines at (a; b). Cubic curves 2 3 2 2 3 C1 = fy − x − x = 0g;C2 = fy − x = 0g have double points at the origin. Indeed, if Ci = fPi(x; y) = 0g then @P @P 1 = −3x2 − 2x; 2 = −3x2; @x @x @P @P 1 = 2y; 2 = 2y; @y @y @2P @2P 1 = −6x − 2; 2 = −6x; @x2 @x2 @2P @2P 1 = 0; 2 = 0; @x@y @x@y @2P @2P 1 = 2; 2 = 2: @y2 @y2 Polynomials (1) are respectively x2 y2 x2 y2 − 2 + 0xy + 2 = (y − x)(y + x); 0 + 0xy + 2 = y2: 2! 2! 2! 2! It follows that only for C1 the double point at the origin is ordinary. 4 4 2 2 2 The curve C3 = f(x + y ) − x y = 0g has a singular point of multiplicity four at 4 4 2 2 2 2 2 the origin which is not ordinary. The curve C4 = f(x + y − x − y ) − 9x y g has an ordinary singular point of multiplicity 4 at the origin. Historically, real algebraic curves were studied first. Let P (x; y) 2 R[x; y] be a non-constant polynomial in 2 variables with real coefficients. A real algebraic curve defined by P is 2 C = f(x; y) 2 R j P (x; y) = 0g: 3 The study of real algebraic curves originates in ancient Greece. Ancient Greeks had sophisticated geometrical methods but a relatively primitive understanding of algebra. To them a circle was not defined by an equation (x − a)2 + (y − b)2 = r2 but it was the locus of all points having equal distance from a fixed point (a; b). Generalising the above point of view we arrive at the definition of conchoid. 2 More precisely, consider a fixed point q 2 R and a fixed constant a 2 R>0. The conchoid of C with respect to q and with parameter a is the locus of all points p 2 R2 such that the line through p and q meets C at a distance a from p. The classical example is the conchoid of Nicomedes (225 B.C.) which is the conchoid of a line with respect to a point not on a line. If the line is x = b and q = (0; 0) is the origin then the conchoid is (x2 + y2)(x − b)2 = a2x2: It can be used to trisect an angle α. Let α be the angle AOB and let D be any point E C A B D L O on jAOj. Line perpendicular to jADj in point D intersects line jOBj in point L. We consider the conchoid of line DL with respect to point O with parameter 2jOLj (i.e. jDEj = 2jOLj). Line parallel to jOAj and passing through L intersect the conchoid in point C. Then the angle AOC is one third of the angle AOB. 4 One can also consider a path of a fixed point on a circle rolling along a line. The obtained curve, cycloid, was discovered by Bernoulli around 1700. It is a path that a particle takes when sliding from one point to another on a vertical plane. This discovery led to the development of the theory of variations. 5 2 19.01. Lecture 2. Complex projective spaces and projective transformations Complex algebraic curves, as defined in the previous lecture, are never compact. For many purposes it is useful to compactify complex algebraic curves by adding points ”infinity". For example, curves 2 2 C1 = fy = x − 1g;C2 = fy = cxg intersect when c 6= ±1. If c = ±1, C1 and C2 do not intersect but they are asymptotic as x and y tend to infinity. We would like to \add points at infinity" to say that C1 and C2 intersect, possibly at infinity, for any value of c 2 C. To do so, we need the notion of a complex projective space P2. Recall that a topological space X is compact if any open cover of X has a finite subcover. A metric space (X; d) is compact if (X; d) is complete (any Cauchy sequence has a limit) and universally bounded (for any " > 0 there exists a finite open cover of X with balls of radius "). X is a Hausdorff space if it satisfies axiom T2, i.e. if for any pair of points x, y of X there exist open subsets U; V ⊂ X such that x 2 U, y 2 V and U \ V = ;.