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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Extra info for A study of singularities on rational curves via syzygies
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