By David Cox, Andrew R. Kustin, Claudia Polini, Bernd Ulrich

Ponder 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 unfastened, 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 element, and the multiplicity of every department. enable 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 observe the overall Lemma to f' for you to 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 concerning the singularities of C within the moment neighbourhood of p. think about rational aircraft curves C of even measure d=2c. The authors classify curves in line 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 research of multiplicity c singularities on, or infinitely close to, a hard and fast rational airplane curve C of measure 2c is resembling 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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**Example text**

Secondly, the parameterization determined by ϕ is birational; that is, ϕ ∈ BHd ; thus, ϕ ∈ M Bal ∩ BHd = M . (2) To read the chart of (3), notice that the chart says for example, that there are 2 multiplicity c singularities on the curve Cc:c,c ; one of these singularities ([0 : 0 : 1]) has an inﬁnitely near singularity of multiplicity c and the other singularity ([1 : 0 : 0]) does not have any inﬁnitely near singularities of multiplicity c. (3) We did not include μ2 in the chart of (3) because the intersection DOBal μ2 ∩BHd , which is also called DOμ2 , is empty.

Notice that D i D ∼ = k [u k) Jac(D/k iD k) = = Jac(D i /k ei −1 D iD . i It follows that k ) = 0 ⇐⇒ Jac(D/k k) = D dim D/ Jac(D/k ⇐⇒ the roots of gcd I3 (A) are distinct and k )) = e(D/ Jac(D/k (ei − 1) and this is equal to the degree of deg gcd I3 (A) minus the number of distinct linear factors of gcd I3 (A). 34 3. 22. 19. It follows that the rings k [T dimension and this common dimension is either 0 or 1; furthermore, these two rings have the same multiplicity. The height of I3 (A) is 2 if and only if the gcd of I3 (A) is a unit.

Bal Proof. By the deﬁnition of DOBal . 21 shows how the matrices Cg and Ag are obtained from C and A . It suﬃces to prove the result when ϕ ∈ M Bal . We have recorded the matrices (C , A ), whose entries are linear forms from T ] and k [u u ], and which satisfy T ϕ = Q1 · · · Qμ C and C u T = A T T , for k [T μ = μ(I1 (ϕ )). The set of linearly independent forms Q1 , . . , Qμ from Bc may be extended to a basis Q1 , . . , Qc+1 for Bc . The entries of ρ(c) also form a basis for Bc ; 4. SINGULARITIES OF MULTIPLICITY EQUAL TO DEGREE DIVIDED BY TWO 45 so there is an invertible matrix υ, with entries in k , so that ρ(c) = [Q1 , .