Download 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

Give some thought to a rational projective curve C of measure d over an algebraically closed box kk. There are n homogeneous types 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 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 each one singular aspect, and the multiplicity of every department. permit p be a unique 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 one can 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 in regards to the singularities of C within the moment neighbourhood of p. think of rational aircraft curves C of even measure d=2c. The authors classify curves in keeping 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 akin to the learn of the scheme of generalised zeros of the mounted balanced Hilbert-Burch matrix f for a parameterisation of C

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Now we prove (2). 11) gives ρ(c) Cλ = T ϕλ . If ht(I1 (Cλ )) ≤ 1, then T ϕλ )) ≤ 1, which shows μ(I1 (Cλ )) ≤ 1 because the entries of C are linear. Thus, μ(T that after row operations ϕλ has at most one non-zero entry. This would imply that ht(I) ≤ 1. 7. 14. 10. The following statements hold. (1) The projections T , u ]) BiProj (kk [T PPP ♥♥ PPP ♥ ♥ ♥ PPP ♥ ♥♥ PPP ♥ ♥ v♥♥ ( T ]) u]) k [T Proj (k Proj (kk [u induce isomorphisms uT ) T , u]/I1 (Cu BiProj k [T ❚❚❚❚ ❥ ❥ ❥ ❚❚❚❚π2 π1 ❥❥❥❥ ❚❚❚❚ ❥❥❥❥∼ ∼ ❚❚❚❚ ❥ = = ❥ ❥ t❥❥ * T ]/I2 (C)) u]/I3 (A)) ; k [T Proj (k Proj (kk [u kT ku T] k [u u] in particular, the schemes Proj( Ik2[T (C) ) and Proj( I3 (A) ) are isomorphic.

Proof. Write Δ = P1 Q2 −P2 Q1 . Notice that gcd(P1 , P2 ) = 1 and gcd(Q3 , Δ) = 1 since I2 (ϕ) has height 2. 9 shows how to modify Q1 and Q2 in order to have gcd(P1 , Q1 ) = 1 and gcd(P2 , Q2 ) = 1. Passing to an affine chart we may assume that p is the origin on the affine curve parametrized by ( gg13 , gg23 ). The blowup of this curve at the origin has two charts, parametrized by ( gg13 , gg21 ) and ( gg23 , gg12 ), respectively. Homogenizing we obtain two curves C and C in P2 parametrized by [g12 : g2 g3 : g1 g3 ] and [g22 : g1 g3 : g2 g3 ], respectively.

Rename Q1 and Q2 : the old Q1 − α4 Q3 becomes the new Q1 and the old Q2 − β4 Q3 becomes the new Q2 . Rename the constants αi and βi . We have transformed ϕ into ⎡ ⎤ Q1 α1 Q1 + α2 Q2 + α3 Q3 ⎣Q2 β1 Q1 + β2 Q2 + β3 Q3 ⎦ . Q3 Q4 Subtract α1 Co1 from Co2 and rename Q4 and β2 . The matrix ϕ has become ⎤ ⎡ α2 Q2 + α3 Q3 Q1 ϕ = ⎣Q2 β1 Q1 + β2 Q2 + β3 Q3 ⎦ . Q3 Q4 At this point there are three cases. Either α3 = β3 = 0 (Case 2A), or α3 = 0 (Case 2B), or α3 = 0 and β3 = 0 (case 2C). 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 .

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