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

Reflect on a rational projective curve C of measure d over an algebraically closed box kk. There are n homogeneous types g1,...,gn of measure d in B=kk[x,y] which parameterise C in a birational, base element loose, demeanour. The authors learn the singularities of C via 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 each one singular element, and the multiplicity of every department. allow p be a unique aspect 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 observe the final Lemma to f' with a view to know 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 profit in regards to the singularities of C within the moment neighbourhood of p. think about 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 examine of multiplicity c singularities on, or infinitely close to, a set rational airplane curve C of measure 2c is similar to the research of the scheme of generalised zeros of the mounted balanced Hilbert-Burch matrix f for a parameterisation of C

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A study of singularities on rational curves via syzygies

Think of a rational projective curve C of measure d over an algebraically closed box kk. There are n homogeneous kinds g1,. .. ,gn of measure d in B=kk[x,y] which parameterise C in a birational, base element unfastened, demeanour. The authors research the singularities of C through learning a Hilbert-Burch matrix f for the row vector [g1,.

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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 . In this case, p1 = [0 : 0 : 1] and p2 = [1 : 0 : 0] are the singularities on C of multiplicity c and there is one singularity of multiplicity c infinitely near to p1 . (6) 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 .

5. In practice we are interested in the geometry which corresponds to BHd . 10 do not require the birationality hypothesis. These results make sense in BalHd . (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.

Let p be the point Ψ(q) on C. 17 (d). 19). 17 (e). 22). 30) is the definition of branch. 31. We say that an ideal of B = k [x, y] is a height one linear ideal if it is generated by one non-zero linear form. There is a one-to-one correspondence between the height one linear ideals of B and the points of P1 . 25 gives a one-to-one correspondence between the height one linear ideals of B and the branches of C. If ( ) is a height one linear ideal of B, then let C( ) be the corresponding branch of C.

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