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

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

Give some thought to 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 aspect unfastened, 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 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 follow the overall Lemma to f' which 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 profit concerning the singularities of C within the moment neighbourhood of p. examine rational airplane 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 airplane curve C of measure 2c is such as 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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In [24] we studied Rees algebras of ideals generated by 3 forms of degree 6 in k [x, y]. We found that there are exactly seven families of Rees algebras for such ideals. Each member of a given family has the same graded Betti numbers. Four of these family arise when every entry in the syzygy matrix has degree 3. The distinguishing invariant turned out to be μ(I2 (C)). Ad hoc methods led to this 3. THE BIPROJ LEMMA 33 classification. 22 that this invariant arises naturally and reflects the geometry of the curve.

10 with c ≥ 2. The following statements hold: (1) ht I2 (C) ≥ 2, (2) ht I3 (A) ≥ 1, and (3) ht I1 (Cλ ) = 3 for general λ ∈ A2 . k [T T ]/I2 (C)) is either empty or is a Proof. 14 shows that Proj(k finite set; therefore, (1) and (2) hold. 7. 20. Let C be matrix of linear forms from a polynomial ring R in three variables over a field k . If C has 2 columns, some generalized column ideal of C is a zero-dimensional ideal of R, and ht(I2 (C)) = 2, then e(R/I2 (C)) + μ(I2 (C)) = 6. Proof. Let R = k [T1 , T2 , T3 ].

12 with A1 = Q1 Q2 A2 = A4 = Q3 and transform ϕ into α2 Q2 β1 Q1 + β2 Q2 A5 = Q4 ⎡ Q1 ⎣Q2 Q3 ⎤ 0 ∗ ⎦, Q4 where Q1 , Q2 , Q3 , Q4 are linearly independent and ∗ is a non-zero element of the vector space . If ∗ ∈ , then ϕ may be transformed into ϕc:c ; otherwise, ϕ may be transformed into ϕc,c . In Case 2B, one may quickly transform α3 into 1. ) At this point, ϕ is ⎡ ⎤ Q1 α2 Q2 + Q3 ⎣Q2 β1 Q1 + β2 Q2 + β3 Q3 ⎦ . Q3 Q4 Add α2 Ro2 to Ro3 and rename Q3 , Q4 , and the β’s to obtain ⎤ ⎡ Q3 Q1 ⎣Q2 β1 Q1 + β2 Q2 + β3 Q3 ⎦ .

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