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 aspect loose, demeanour. The authors examine 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 element, and the multiplicity of every department. permit p be a novel aspect 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' with the intention 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 concerning the singularities of C within the moment neighbourhood of p. think about rational airplane 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 learn of multiplicity c singularities on, or infinitely close to, a set rational airplane curve C of measure 2c is corresponding 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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Additional info for A study of singularities on rational curves via syzygies
Finally, if μ = 2, then one quickly puts ϕ in the form ⎡ ⎤ ∗ ∗ ⎣∗ ∗ ⎦ . 0 Q2 Further elementary row and column operations transform ϕ into ⎡ ⎤ ⎡ ⎤ Q1 αQ1 0 Q1 ⎣Q2 βQ1 ⎦ , then ⎣Q2 βQ1 ⎦ , 0 Q2 0 Q2 for some constants α and β. The constant β is non-zero since ht I2 (ϕ) = 2 and there exists g ∈ G with gϕ ∈ MμBal . We have shown that the BalHd = ∪ ∈ECP DOBal . 10 shows that the orbits DOBal , with ∈ ECP, are disjoint. 4. 10. 12) with ϕ ∈ DOBal . (1) One may transform the matrices (C , A ), using elementary operations and the suppression of zero rows, into the matrices (C , A ), which are given by: ⎡ T1 T2 T3 0 0 0 0 0 0 T1 T2 T3 C(c,μ5 ) = T1 T2 0 0 0 0 0 T1 T2 T3 C(c,μ4 ) = T1 T2 0 0 T2 0 T1 T3 C(∅,μ6 ) =⎣ Cc:c = Cc:c,c = C μ2 = T1 T2 0 0 ⎤ ⎡ u1 0 u1 0 0 u2 0 0 0 u1 0 0 u2 , A(c,μ5 ) = u1 0 u2 0 0 0 u1 0 u2 0 0 0 0 0 u2 , C(∅,μ4 ) = , A(c,μ4 ) = u1 0 u2 0 u2 u1 0 0 0 0 0 u2 , Cc,c = ⎦ , A(∅,μ ) = ⎣ 6 0 T3 T1 T2 T1 0 T2 T3 0 T2 T1 T2 T2 T3 ⎤ 0 0 u2 0 0 , Ac:c = , Ac:c,c = , A μ2 = u1 0 u2 0 0 u1 0 u2 0 u2 0 0 , Cc:c:c = u1 0 0 0 u1 u2 0 u2 0 u1 u2 0 0 u1 u2 ⎦ , C(∅,μ ) = 5 , Cc,c,c = T1 T2 T3 0 0 T1 T2 T3 0 T1 0 T2 0 0 T1 T2 T3 T1 T2 T2 T3 0 T1 T1 T1 T2 0 0 T3 T3 0 0 T1 T2 0 T1 T2 T3 , A(∅,μ5 ) = u1 0 0 u2 0 u1 u2 0 0 0 u1 0 0 u2 u2 0 u1 0 0 , 0 u1 u2 0 0 0 u1 u2 , , Ac,c = u1 0 0 u2 0 (u1 +u2 ) 0 0 0 0 0 u2 , , Ac:c:c = u1 u2 0 0 u1 u2 u2 0 0 , A(∅,μ4 ) = , Ac,c,c = , u1 +u2 0 0 0 u1 0 0 0 u2 , .
We ﬁrst prove that there exists g ∈ G with gϕ ∈ M Bal for some ∈ ECP. Consider the parameter μ = μ(I1 (ϕ)). The matrix ϕ has six homogeneous entries of degree c, so μ ≤ 6. On the other hand, the hypothesis that ht I2 (ϕ) = 2 guarantees that 2 ≤ μ. Thus, 2 ≤ μ ≤ 6. We treat each possible value for μ separately. If μ = 6, then the entries of ϕ are linearly independent and Bal . Suppose now that μ = 5. 7), then, after row and column operations, ϕ is transformed into gϕ ∈ M(c,μ . If 5) ϕ does not have a generalized zero, one may apply row and column operations to put ϕ in the form ⎡ ⎤ Q1 Q4 ⎢Q2 Q5 ⎥ ⎢ ⎥, 5 ⎣ ⎦ Q3 αi Qi i=1 where the αi ∈ k are constants.
To ﬁnd a singularity of multiplicity c on C we need to describe a generalized zero of ϕ. In other words, we look for (p, q) in P2 × P1 such that uT ∈ k [T T , u , x, y], where T = [T1 , T2 , T3 ] pϕq T = 0. Consider the polynomial T ϕu and u = [u1 , u2 ] are matrices of indeterminates. We extract the variables x and uT . 1) ρ(i) = [y i , xy i−1 , x2 y i−2 , . . , xi−1 y, xi ]. Deﬁne C and A to be the matrices with T ϕ = ρ(c) C uT = AT T T, and Cu T ] and the entries of A are linear forms so that the entries of C are linear forms in k [T u].