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

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 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. permit p be a novel 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 observe 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. give some thought to rational aircraft curves C of even measure d=2c. The authors classify curves based on 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 similar to the examine of the scheme of generalised zeros of the fastened balanced Hilbert-Burch matrix f for a parameterisation of C

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Notice that ∞ JSm : I1 (yy )∞ = y1 , . . , yn−1 , I1 (φ )yn Sm : y1 , . . , yn−1 , yn = y1 , . . , yn−1 , I1 (φ )yn Sm : (yn )∞ = y1 , . . , yn−1 , I1 (φ ) Sm . 3. THE BIPROJ LEMMA 29 Therefore, the image of π| is deﬁned scheme-theoretically by (JSm : I1 (yy )∞ ) ∩ Rm = I1 (φ )Rm = In (φ)Rm , which proves the second assertion. To show the ﬁrst claim, notice that the natural map Sm Rm [yn ] = Rm → I1 (φ )yn Rm [yn ] JSm induces π| : Proj(Sm /JSm ) → Spec(Rm ); furthermore, Sm JSm Rm [yn ] Rm [yn ] = Proj I1 (φ )yn Rm [yn ] I1 (φ )yn Rm [yn ] : (yn )∞ Rm [yn ] Rm Rm = Proj [y ] = Spec = Proj I1 (φ )Rm [yn ] I1 (φ )Rm n I1 (φ )Rm Rm = Spec .

We ﬁrst show that if q is a singularity of multiplicity c inﬁnitely near to C, then q is inﬁnitely near to a singularity p on C of multiplicity c. In particular, it is not possible for c = mq < mp . The most signiﬁcant part of the result is assertion (4), where we study singular points of multiplicity c which are in the second neighborhood of C. 8. Remark. Assertion (1) in the next result is a purely geometric statement. 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.

THE GENERAL LEMMA 21 We translate the information we have collected about the rings R ⊆ T and ˆ R ⊆ Tˆ to information about the rings OC,p ⊆ OC,p and OC,p ⊆ OC,p . Recall k [Id ]) = Quot(k k [Bd ]): ﬁrst that all four rings are subrings of Quot(k OC,p = { fg | f ∈ k [Id ] and g ∈ k [Id ] \ p are homogeneous of the same degree}, OC,p = { fg | f ∈ k [Bd ] and g ∈ k [Id ] \ p are homogeneous of the same degree}, R = k [Id ]p = { fg | f ∈ k [Id ] and g ∈ k [Id ] \ p}, and T = k [Bd ]p = { fg | f ∈ k [Bd ] and g ∈ k [Id ] \ p}.