Chapter 7 Local properties of plane algebraic curves - RISC

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Theorem 7.3.4. Let the curve F of degree n have no multiple components and let P 1 , . . .,P m be the singularities of F with multiplicities r 1 , . . .,r m , respectively. Then n(n − 1) ≥ m∑ r i (r i − 1). i=1 Proof: Choose the coordinate system such that the point (0 : 0 : 1) is not contained in F (i.e. the term z n occurs with non-zero coefficient in F). Then F z (x, y, z) := ∂F ∂z ≠ 0. Moreover, F does not have a factor independent of z. Since F is squarefree, F can have no factor in common with F z . The curve F z has every point P i as a point of multiplicity at least r i −1, since every j–th derivative of F z is a (j + 1)-st derivative of F. Thus, by application of Theorem 7.3.3 we get the result. This bound can be sharpened even more for irreducible curves. Theorem 7.3.5. Let the irreducible curve F of degree n have the singularities P 1 , . . .,P m of multiplicities r 1 , . . ., r m , respectively. Then (n − 1)(n − 2) ≥ m∑ r i (r i − 1). i=1 Proof: From Theorem 7.3.4 we get ∑ ri (r i − 1) ≤ 2 n(n − 1) 2 ≤ (n − 1)(n + 2) . 2 (Note: one can always determine a curve of degree n − 1 satisfying (n − 1)(n + 2)/2 linear conditions. By r(r − 1)/2 linear conditions we can force a point P to be an (r − 1)–fold point on a curve.) So, there is a curve G of degree n − 1 having P i as (r i − 1)–fold point, 1 ≤ i ≤ m, and additionally passing through (n − 1)(n + 2) 2 − ∑ ri (r i − 1) simple points of F. Since F is irreducible and deg(G) < deg(F), the curves F and G can have no common component. Therefore, by Theorem 7.3.3, 2 n(n − 1) ≥ ∑ r i (r i − 1) + (n − 1)(n + 2) 2 − ∑ ri (r i − 1) , 2 108
and consequently (n − 1)(n − 2) ≥ ∑ r i (r i − 1). Indeed, the bound in Theorem 7.3.5 is sharp, i.e. it is actually achieved by a certain class of irreducible curves. These are the curves of genus 0. Definition 7.3.4. Let F be an irreducible curve of degree n, having only ordinary singularities of multiplicities r 1 , . . .,r m . The genus of F, genus(F), is defined as genus(F) = 1 m [(n − 1)(n − 2) − 2 ∑ r i (r i − 1)]. i=1 Example 7.3.4. Every irreducible conic has genus 0. An irreducible cubic has genus 0 if and only if it has a double point. For curves with non-ordinary singularities the formula m 1 2 [(n − 1)(n − 2) − ∑ r i (r i − 1)] is just an upper bound for the genus. Compare the tacnode curve of Example 1.3, which is a curve of genus 0. We will see this in the chapter on parametrization. i=1 109
Page 1 and 2: Chapter 7 Local properties of plane
Page 3 and 4: irreducible factors of the polynomi
Page 5 and 6: So the tangent to F at P is defined
Page 7 and 8: Every point (a, b) on C corresponds
Page 9 and 10: Now, we proceed to determine and an
Page 11 and 12: 7.2 Intersection of Curves In this
Page 13 and 14: of C and D. But this implies that t
Page 15 and 16: factor D(z) intersection point mult
Page 17 and 18: From Theorem 7.2.3 one can extract
Page 19 and 20: mult O (e ∩ y) = (4),(7) mult O (
Page 21 and 22: F 0 and F 1 determine a 1-dimension
Page 23: component, which must be G, since G

Chapter 7 Local properties of plane algebraic curves - RISC

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