# A study of singularities on rational curves via syzygies by David Cox, Andrew R. Kustin, Claudia Polini, Bernd Ulrich

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 kinds 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 by way 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 every one singular aspect, 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 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 practice the overall Lemma to f' so that it will 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 benefit in regards to the singularities of C within the moment neighbourhood of p. examine 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 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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**A study of singularities on rational curves via syzygies**

Reflect on a rational projective curve C of measure d over an algebraically closed box kk. There are n homogeneous kinds g1,. .. ,gn of measure d in B=kk[x,y] which parameterise C in a birational, base element loose, demeanour. The authors examine the singularities of C by way of learning a Hilbert-Burch matrix f for the row vector [g1,.

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7. The ring extension k [Id ] ⊆ B is integral; so every maximal ideal of S has the form qS, where q is a height one homogeneous prime ideal of B which is minimal over pB and which satisﬁes q ∩ k [Id ] = p. Let q be a homogeneous prime ideal in B for which qS is a maximal ideal of S. The ideal q is principal and is generated by some homogeneous form f ∈ B. Let q ∈ Pk1¯ be a root of f . 17 (c), are in the ideal q = (f ); and therefore, the generators of p all vanish at q. It follows −1 (p) ensures that q is already in P1 ; that Ψk¯ (q) = p.

8 will eventually lead to a better understanding of the other singularities on and inﬁnitely near the curve C. An analysis of this sort for quartics is carried out in Chapter 8. A future paper will contain our analysis of all singularities on sextics. Let C be a rational plane curve of degree d = 2c. 5, that if there is a multiplicity c singularity on, or inﬁnitely near, C, then every entry in a homogeneous Hilbert-Burch matrix for a parameterization of C is a form of degree c. 3) into 11 disjoint orbits under the action k )×GL2 (k k ).

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 ). In the ﬁrst application R = k [T T ]. 13 ensures that I1 (C) is = I1 (AT T ]. In the second application R = k [u u ]. 13 zero-dimensional in k [T u]. Assertion (1) is established. ensures that I2 (A) is zero-dimensional in k [u Return to the ﬁrst setting. 5 also yields that the image of the map π k [T T ]) P2 × P1 ⊇ BiProj(S/J) − → Proj(k 3. THE BIPROJ LEMMA 31 k [T T ]/I2 (C)) ⊆ Proj(k k [T T ]).