3 Tangent spaces, dimension and singulari- ties n Let H ⊂ A (K) be a hypersurface defined by the equation f(x1; x2; : : : ; xn) = 0, where f 2 K[x1; x2; : : : ; xn], and and let P 2 H. One of the possible ways n of defining that the line fP + tv j t 2 Kg (v = (v1; v2; : : : ; vn) 2 K n f0g) is tangent to H at P is to require that f(P + tv), now simply a function of t, has a stationary point at t = 0. The Taylor expansion of f(P + tv) about t = 0 is df(P + tv) f(P + tv) = f(P ) + t +(terms of order ≥ 2 in t) dt P n X @ f = f(P ) + t vj + (terms of order ≥ 2 in t). @ x P j=1 j (Derivatives of polynomials can be defined purely formally over any field without using the concept of limits and Taylor expansion works for polyno- mials over any field.) n X @f Thus f(P +tv) has a stationary point at t = 0 if and only if vj = 0. @x P j=1 j n @f @f @f X @f By using the gradient rf = ; ; ··· ; , the condition vj = @x @x @x @x P 1 2 n j=1 j 0 can be re-written as (rf)P :v = 0. Definition. Let V ⊆ An(K) be an affine algebraic variety and let P 2 V .A line ` given in parametric form as fP + tv j t 2 Kg,(v 2 Kn n f0g), is said to be tangent to V at P if and only if (rf)P :v = 0 for every f 2 I(V ). Definition. The tangent space to V at P , denoted by TP V is the union of the tangent lines to V at P and the point P . n Theorem 3.1 Let V ⊆ A be an affine algebraic variety. Let f1, f2,..., fr be a set of generators for I(V ). Let P 2 V and let v = (v1; v2; : : : ; vn) 2 Kn n f0g. The line ` = fP + tv j t 2 Kg is tangent to V at P if and only if 33 JP v = 0, where J is the Jacobian matrix, 0 @f @f @f 1 1 1 ··· 1 B@x1 @x2 @xn C B C B @f2 @f2 @f2 C B ··· C J = B@x1 @x2 @xn C B . C B . .. C B C B C @ @f @f @f A r r ··· r @x1 @x2 @xn and JP is J evaluated at P . Proof. Assume that ` is tangent to V at P . Then (rf)P :v = 0 for every f 2 I(V ), in particular, (rfi)P :v = 0 for every i, 1 ≤ i ≤ r.(rfi)P :v is exactly the ith component of JP v, therefore JP v = 0. r P Assume now that JP v = 0. Let f 2 I(V ). Then f = gifi for some i=1 gi 2 K[x1; x2; : : : ; xn], 1 ≤ i ≤ r. Now r r X X rf = r(gifi) = ((rgi)fi + girfi); i=1 i=1 therefore r r X X (rf)P = ((rgi)P fi(P ) + gi(P )(rfi)P ) = gi(P )(rfi)P : i=1 i=1 In the last step we used that fi(P ) = 0 for every i, 1 ≤ i ≤ r. Hence r P (rf)P :v = gi(P )(rfi)P :v, but (rfi)P :v = 0, because it is the ith com- i=1 ponent of JP v = 0, therefore (rf)P :v = 0. n It follows from this theorem that the set W = fv 2 K j JP v = 0g is a linear subspace, therefore TP V = P + W is an affine subspace. Its dimension is dim TP V = dim W = n − rank JP by the Rank-Nullity Theorem. Example: Let V ⊂ A2(C) be the cuspidal cubic curve defined by the equation y2 − x3 = 0. It can be checked that y2 − x3 is irreducible, so hy2 − x3i is a prime ideal, therefore it is radical and then the Nullstellensatz implies that that I(V ) = hy2 − x3i. Therefore J = (−3x3 2y). 34 y 2 1 x -0.5 0.5 1.0 1.5 2.0 2.5 -1 -2 At P = (1; 1), JP = (−3 2), therefore the line fP + tv j t 2 Cg is tangent to V at (1; 1) if and only if −3v1 + 2v2 = 0. Hence T(1;1)V is the line of slope 3=2 through P with equation y = 3x=2 − 1=2. It agrees with the tangent line obtained by, for example, implicit differentiation. At Q = (0; 0), however, something strange happens. JQ = (0; 0), therefore 2 every line through (0; 0) is a tangent line and T(0;0)V = A ! Let v = (α; β) 6= (0; 0) be the direction vector of a line through Q = (0; 0). Then Q + tv = (tα; tβ) and if we substitute x = tα, y = tβ into y2 − x3 we obtain t2β2 − t3α3 = t2(β2 − tα3), which has a zero of multiplicity at least 2 at t = 0, so any line through (0; 0) is a tangent line by our definition. If β = 0, then t2(β2 − tα3) has a zero of multiplicity 3 at t = 0, so the line y = 0 has a higher order tangency to the curve than a general line through (0; 0). Proposition–Definition 3.2 Let V ⊆ An be an irreducible affine algebraic variety. If V 6= ;, there exists a proper subvariety Z of V (Z 6= V , but Z may be ;) and an integer d ≥ 0 such that dim TP V = d for every P 2 V n Z and dim TP V > d for P 2 Z. (In other words, dim TP V is constant on a non- empty Zariski open subset of V .) The dimension of V , denoted by dim V , is defined to be d. The points of V nZ, where dim TP V = dim V , are called non- singular, while the points of Z, where dim TP V > dim V , are called singular and the set of singular points of V is denoted by Sing V . (Sometimes it is convenient to define dim ; = −1 and Sing ; = ;.) Proof. Let Vm = fP 2 V j dim TP V ≥ mg, where m ≥ 0 is an integer. By the remark after Theorem 3.1, Vm = fP 2 V j rank JP ≤ n − mg. It is a theorem in linear algebra that a matrix has rank ≤ k if and only if all of its 35 (k + 1) × (k + 1) minors vanish. Therefore Vm = V \V(h(n − m + 1) × (n − m + 1) minors of Jg: This shows that Vm is the affine algebraic variety defined by the ideal gener- ated by I(V ) and by the (n − m + 1) × (n − m + 1) minors of J. We have V = V0 ⊇ V1 ⊇ V2 ⊇ ::: ⊇ Vn ⊇ Vn+1 = ;. Choose d maximal such that Vd = V . Then dim TP V = d for P 2 V nVd+1 and dim TP V > d for P 2 Vd+1, so dim V = d and Sing V = Vd+1. n Example: If V is an affine subspace of A , TP V = V at every point P 2 V , so the dimension of V as a variety agrees with its dimension as an affine space and it has no singular points. In particular, An is non-singular and has dimension n. Worked examples can be found in the separate handout at https://personalpages. manchester.ac.uk/staff/gabor.megyesi/teaching/MATH32062/singularities. pdf. Herwig Hauser's gallery at http://homepage.univie.ac.at/herwig. hauser/bildergalerie/gallery.html has many nice examples of singular surfaces. The combination of the preceding results gives the following procedure for finding the dimension and singular points of an irreducible variety V ⊆ An(K). 1. Choose a set of generators f1; f2; : : : ; fr for I(V ). (In all the problems you will have to solve, the field will be algebraically closed and the defining ideal of V will be a radical ideal, so it will be equal to I(V ).) 2. Calculate the Jacobian matrix J. 3. For k = 0; 1; 2;:::, find the points where all the (k + 1) × (k + 1) minors of J vanish and also f1 = f2 = ::: = fr = 0. These are the points where rank JP ≤ k and therefore dim TP V ≥ n − k. Do this until you find the smallest k such that dim TP V ≥ n − k for every point P 2 V . 4. Then dim V = n−k and Sing V is the set of points where dim TP V > n−k, i. e., the points of V where all the k × k minors of J vanish. Definition. Let V be an arbitrary (not necessarily irreducible) affine algebraic variety. The dimension of V , dim V , is the maximum of the dimensions of the irreducible components of V . The local dimension of V at P 2 V is defined to be the maximum of the dimension of the irreducible components of V containing P . P is called non-singular if and only if dim TP V is equal to the local dimension of V at P . P is called singular if and only if dim TP V is greater than the local dimension 36 of V at P . The set of singular points of V is denoted by Sing V , just as in the irreducible case. Fact: Sing V consists of the singular points of the individual irreducible com- ponents and of intersection points of different components. This means that in general, one needs to decompose the variety into its irreducible components in order to find the its dimension and its singular points, however, by the proposition below and by question 2 on tutorial sheet 7, hypersurfaces over an algebraically closed field are an exception.