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

Think about 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 aspect loose, demeanour. The authors research 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 every one singular aspect, and the multiplicity of every department. allow 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 supply 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' in an effort to 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. give some thought to 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 learn of multiplicity c singularities on, or infinitely close to, a set rational aircraft curve C of measure 2c is comparable to the examine of the scheme of generalised zeros of the fastened balanced Hilbert-Burch matrix f for a parameterisation of C

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Q3 Q4 48 4. SINGULARITIES OF MULTIPLICITY EQUAL TO DEGREE DIVIDED BY TWO Subtract β2 Co1 from Co2 , β2 Ro1 from Ro3 , and rename Q3 and Q4 to obtain ⎡ ⎤ Q1 Q3 ⎣Q2 β1 Q1 ⎦ . Q3 Q4 If β1 is zero, then ϕ may be transformed into ϕ(c,μ4 ) . If β1 is not zero, then β1 may be transformed into 1. ) At this point one uses row and column exchanges to transform ϕ into the form of ϕ(∅,μ4 ) . In Case 2C, one starts with ⎤ ⎡ α2 Q2 Q1 ϕ = ⎣Q2 β1 Q1 + β2 Q2 + β3 Q3 ⎦ , Q3 Q4 with β3 = 0. Transform β3 into 1 by multiplying Co2 by β3−1 and renaming Q4 and the constants.

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. kT T] (2) As a subset of P2 , Proj( Ik2[T (C) ) is equal to {p ∈ C | mp = c}. T , u ] and J = I1 (Cu uT ) Proof. 5 twice. Each time S = k [T T T ).

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.

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