Download A study of singularities on rational curves via syzygies by David Cox, Andrew R. Kustin, Claudia Polini, Bernd Ulrich PDF

By David Cox, Andrew R. Kustin, Claudia Polini, Bernd Ulrich

ISBN-10: 0821887432

ISBN-13: 9780821887431

Examine a rational projective curve C of measure d over an algebraically closed box kk. There are n homogeneous varieties g1,...,gn of measure d in B=kk[x,y] which parameterise C in a birational, base element unfastened, demeanour. The authors learn the singularities of C by way of learning a Hilbert-Burch matrix f for the row vector [g1,...,gn]. within the ""General Lemma"" the authors use the generalised row beliefs of f to spot the singular issues on C, their multiplicities, the variety of branches at every one singular element, and the multiplicity of every department. permit p be a unique element at the parameterised planar curve C which corresponds to a generalised 0 of f. within the ""Triple Lemma"" the authors provide a matrix f' whose maximal minors parameterise the closure, in P2, of the blow-up at p of C in a neighbourhood of p. The authors practice the final Lemma to f' so as to find out about the singularities of C within the first neighbourhood of p. If C has even measure d=2c and the multiplicity of C at p is the same as c, then he applies the Triple Lemma back to benefit concerning the singularities of C within the moment neighbourhood of p. reflect on rational airplane curves C of even measure d=2c. The authors classify curves in keeping with the configuration of multiplicity c singularities on or infinitely close to C. There are 7 attainable configurations of such singularities. They classify the Hilbert-Burch matrix which corresponds to every configuration. The research of multiplicity c singularities on, or infinitely close to, a hard and fast rational aircraft curve C of measure 2c is akin to the learn of the scheme of generalised zeros of the mounted balanced Hilbert-Burch matrix f for a parameterisation of C

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We wonder if there is a geometric argument for it. 5. 1 with n = 3, k an algebraically closed field, and d equal to the even number 2c. (1) If p is a point on C and q is a singularity of multiplicity c infinitely near to p, then the multiplicity of p is also equal to c. (2) If p is a singularity on C of multiplicity c, then the parameters d1 and d2 are both equal to c. (3) Suppose that p is a singularity on C of multiplicity c. Then there is a singularity of C, infinitely near to p, of multiplicity c if and only if there exist invertible matrices χ and ξ, with entries in k , such that (0, 0, 1) = pχ and ⎤ ⎡ P1 Q1 χϕξ = ⎣Q3 Q2 ⎦ , 0 Q3 where the homogeneous forms P1 and Qi all have degree c.

5) The action of G on Hd restricts to given actions of G on BalHd and also on BHd . 9. (6) The well known formula ξ −1 = (det ξ)−1 Adj ξ, where Adj ξ is the classical adjoint of ξ, expresses the inverse of the matrix as a rational function in the entries of ξ. Thus, the function Υ : G × Hd → Hd , which is defined by Υ(g, ϕ) = gϕ, is a morphism of varieties. There are 7 configurations of multiplicity c singularities in CP, but 11 disjoint orbits in our decomposition of BalHd . We form the Total Configuration Poset (TCP) by adjoining 6 new elements to CP and the Extended Configuration Poset (ECP) by removing 2 elements of CP from TCP.

4) If the configuration of multiplicity c singularities on or infinitely near C is described by {c : c}, then ΛC is parameterized by the signed maximal order minors of ϕ ∈ Mc:c . In this case, p = [0 : 0 : 1] is the singularity on C of multiplicity c and there is one singularity of multiplicity c infinitely near to p. (5) If the configuration of multiplicity c singularities on or infinitely near C is described by {c : c, c}, then ΛC is parameterized by the signed maximal order minors of ϕ ∈ Mc:c,c .

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