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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**Example text**

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.