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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

Think about 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 examine 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 aspect, and the multiplicity of every department. permit 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 practice the overall Lemma to f' so that it will 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. reflect on rational aircraft curves C of even measure d=2c. The authors classify curves in response to 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 airplane curve C of measure 2c is comparable 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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7. The ring extension k [Id ] ⊆ B is integral; so every maximal ideal of S has the form qS, where q is a height one homogeneous prime ideal of B which is minimal over pB and which satisfies q ∩ k [Id ] = p. Let q be a homogeneous prime ideal in B for which qS is a maximal ideal of S. The ideal q is principal and is generated by some homogeneous form f ∈ B. Let q ∈ Pk1¯ be a root of f . 17 (c), are in the ideal q = (f ); and therefore, the generators of p all vanish at q. It follows −1 (p) ensures that q is already in P1 ; that Ψk¯ (q) = p.

The ring OC,p is a subring of T . 22) → aT, ← A, provide a one-to-one correspondence between the maximal ideals a of OC,p and the maximal ideals A ∈ {Mi ∩ T | 1 ≤ i ≤ v} of T . Recall that the invariant sC,p is equal to the number of branches of C at p. Thus, sC,p = the number of minimal prime ideals of OC,p = the number of maximal prime ideals of OC,p = the number of maximal prime ideals of T = v. ˆ i ) and e(OC,p /Ji ), where Ji is the Finally, we compare the multiplicities e(R/J ˆ We minimal prime of OC,p which corresponds to the minimal prime ideal Ji of R.

The curve itself might have 0, 1, 2, or 3 singularities of multiplicity c and any one of these singularities has 0, 1, or 2 infinitely near singularities of 37 38 4. SINGULARITIES OF MULTIPLICITY EQUAL TO DEGREE DIVIDED BY TWO multiplicity c, provided that the total number is at most 3. The seven possibilities are: ∅, {c}, {c, c}, {c, c, c}, {c : c}, {c : c, c}, and {c : c : c}, where a colon indicates an infinitely near singularity, a comma indicates a different singularity on the curve, and ∅ indicates that there are no singularities of multiplicity c on, or infinitely near, C.

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

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