By David Cox, Andrew R. Kustin, Claudia Polini, Bernd Ulrich
Contemplate 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 loose, demeanour. The authors learn the singularities of C through 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 aspect, 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 follow the final Lemma to f' for you 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 profit 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 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 research of multiplicity c singularities on, or infinitely close to, a hard and fast rational airplane curve C of measure 2c is akin 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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Extra info for A study of singularities on rational curves via syzygies
To prove (2), one quickly calculates μ(I2 (C )) for each matrix C of assertion (1). It is clear that I2 (C ) = I2 (C ). One also calculates gcd I3 (A ) = gcd I3 (A ): (∅, μ6 ) (∅, μ5 ) (c, μ5 ) (∅, μ4 ) (c, μ4 ) c, c gcd I3 (A ) 1 c:c 1 c:c:c u2 c : c, c 1 c, c, c u2 μ2 u2 (u1 + u2 ). 21 shows that the transformation A to Ag−1 , for g ∈ G, replaces u1 and u]. 22 u2 with linearly independent linear forms 1 and 2 from k [u guarantees that there are exactly 6 − μ(I2 (C )) distinct singularities of multiplicity c on or inﬁnitely near C .
One now has in k [u uT = ρ(c) Cu uT = ρ(c) AT T T. 2) the set of (p, q) in P2 × P1 such that pϕq T = 0 is the zero set, in P2 × P1 , uT ) = I1 (AT T T ). of the bihomogeneous ideal I1 (Cu 27 28 3. 22 to describe the singularities on C of multiplicity c. 4) TT uT = AT Cu provides symmetry. 3). 5. Let S = k [x1 , . . , xm , y1 , . . , yn ] be a bi-graded polynomial ring with deg xi = (1, 0) and deg yi = (0, 1), and R be the sub-algebra k [x1 , . . , xm ] of S. Let J be an S-ideal generated by bi-homogeneous forms which are linear in the y’s.
21 shows that the transformation A to Ag−1 , for g ∈ G, replaces u1 and u]. 22 u2 with linearly independent linear forms 1 and 2 from k [u guarantees that there are exactly 6 − μ(I2 (C )) distinct singularities of multiplicity c on or inﬁnitely near C . 5 to identify these singularities. The following small calculation provides a suﬃcient condition for the existence of a generalized zero in a matrix ϕ. 10. The most important application of this calculation occurs when the parameters “b” and “N ” are both taken to be zero.