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

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 kinds g1,...,gn of measure d in B=kk[x,y] which parameterise C in a birational, base element loose, demeanour. The authors research the singularities of C by means 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. allow p be a novel aspect at the parameterised planar curve C which corresponds to a generalised 0 of f. within the ""Triple Lemma"" the authors supply 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 follow the final Lemma to f' for you 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 benefit in regards to the singularities of C within the moment neighbourhood of p. examine rational airplane curves C of even measure d=2c. The authors classify curves in accordance 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 learn of multiplicity c singularities on, or infinitely close to, a hard and fast rational airplane curve C of measure 2c is comparable to the learn of the scheme of generalised zeros of the fastened balanced Hilbert-Burch matrix f for a parameterisation of C

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THE BIPROJ LEMMA into one of the following four forms: ⎡ T1 M1 = ⎣T2 T3 ⎤ ∗ ∗⎦ , ∗ ⎡ T1 ⎢ T2 ⎢ M2 = ⎣ T3 0 ⎤ f1 f2 ⎥ ⎥, f3 ⎦ T1 ⎡ T1 ⎢ T2 ⎢ M3 = ⎢ ⎢ T3 ⎣0 0 ⎤ g1 g2 ⎥ ⎥ g3 ⎥ ⎥, T1 ⎦ T2 ⎡ T1 ⎢ T2 ⎢ ⎢ T3 M4 = ⎢ ⎢0 ⎢ ⎣0 0 ⎤ 0 0⎥ ⎥ 0⎥ ⎥, T1 ⎥ ⎥ T2 ⎦ T3 where the fi are linear forms in k [T2 , T3 ] and the gi are linear forms in R = k [T3 ]. In the case C = M1 , the ideal I2 (C) is a perfect ideal of height 2 with a 2-linear resolution; hence e(R/I2 (C)) = 3 = μ(I2 (C)). In the case C = M2 , we have I2 (C) = T1 (T1 , T2 , T3 )+JR, where J is a non-zero ideal of k [T2 , T3 ] generated by quadrics.

We call ∗3 by the name Q3 and we have Q1 , Q2 , Q3 linearly independent. At least one of the entries ∗1 or ∗2 is not in . Apply a row exchange and rename Q1 and Q2 , if necessary. We have transformed ϕ into the form ⎡ ⎤ Q1 α3 Q3 + α4 Q4 ⎣Q2 ⎦, Q4 0 Q3 with Q1 , Q2 , Q3 , Q4 linearly independent and α3 , α4 in k . Let Roi and Coi represent row i and column i, respectively. Subtract α4 Ro2 +α3 Ro3 from Ro1 and rename Q1 − α4 Q2 as Q1 . We have transformed ϕ into ⎤ ⎡ Q1 0 ϕ = ⎣Q2 Q4 ⎦ , 0 Q3 which can be transformed into the form of ϕc,c .

21) The ideal pTi is completely determined by the (qi ∩ k [Bd ])-primary component of k [Bd ]. We have already computed the primary components of pB corresponding pk to the prime ideals minimal in the support of B/pB: pB : (x, y)∞ = qc11 ∩ · · · ∩ qcss . The rings k [Bd ]qi ∩kk [Bd ] ⊆ Bqi are DVRs. One can choose the same generator for the maximal ideal of these two rings. There is no difficulty in seeing that if f is an element of k [Bd ] and r is arbitrary, then f ∈ (qi ∩ k [Bd ])rk [Bd ]qi ∩kk [Bd ] ⇐⇒ f ∈ qri Bqi ; k [Bd ])-primary component and therefore, there is no difficulty in seeing that the (qi ∩k k [Bd ] is (qi ∩ k [Bd ])ci .

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