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As Jack Huizenga suggested, in general perhaps one obtains the canonical blowup of W(r,d) by repeatedly blowing up in the usual sense along subvarieties W(s,d) with s > r, or their total transforms, and finally blowing down partially once again.

As Jack Huizenga suggested, in general perhaps one obtains the canonical blowup of W(r,d) by repeatedly blowing up in the usual sense along subvarieties W(s,d) with s > r, or their total transforms, and finally blowing down partially once.

This is an attempt to provide examples and further comments on the nice answer of Jack Huizenga to question 2. I apologize if this attempt is more naive than the question and its previous answers.

Briefly, the “canonical blowup” G(r,d) of W(r,d), (although an abstract blowup in Grothendieck’s sense of some unknown sheaf of ideals), is not a blowup in the usual sense of blowing up a specific closed reduced subvariety such as W(r+1,d), by inserting the projectivized normal cone. It is rather only a naturally computable birational “incidence” map, which is however sometimes a preliminary tool to compute the usual blowup. The usefulness of the “canonical blowup” is that it is specifically computable, birational, and sometimes smooth. Thus knowing G(r,d) may allow one to calculate the usual geometric blowup of W(r,d) at a point.

This is an attempt to provide examples and further comments on the nice answer of Jack Huizenga to question 2. I apologize if this attempt is more naive than the question and its previous answers. The interest of the question is signaled by the fact that already for r = 0, the relation between the canonical blowup and the usual blowup of W(0,d) at one point, is the content of the Riemann - Kempf singularity theorem.

As pointed out above, the “canonical blowup” G(r,d) of W(r,d), (although an abstract blowup in Grothendieck’s sense of some unknown sheaf of ideals), is not a blowup in the usual sense of blowing up a specific closed reduced subvariety such as W(r+1,d), by inserting the projectivized normal cone. It is rather only a naturally computable birational “incidence” map, which is however sometimes a preliminary tool to compute the usual blowup. The usefulness of the “canonical blowup”, it seems to me, is that it is specifically computable, birational, and sometimes smooth. Thus knowing G(r,d) may allow one to calculate the usual geometric blowup of W(r,d) at a point.

This is an attempt to provide examples and further comments on the nice answer of Jack Huizenga to question 2.

Note moreover that the fiber of the usual blowup of W(0,3) at a point L with h^0(L) = 2, is the unique quadric surface in the P^3 containing the canonical curve C, and the two Abel fibers |L| and |K-L| are the systems of divisors cut by the two rulings of that quadric. Thus the two fibers of the “canonical blowup” of W(0,3) over the points L and K-L, correspond to blowing down the fiber of the usual blowup along opposite rulings to obtain two different copies of P^1.

This is an attempt to provide examples and further comments on the nice answer of Jack Huizenga to question 2. I apologize if this attempt is more naive than the question and its previous answers.

Note moreover that the fiber of the usual blowup of W(0,3) at a point L with h^0(L) = 2, is the unique quadric surface in the P^3 containing the canonical curve C, and the two Abel fibers |L| and |K-L| are the systems of divisors cut by the two rulings of that quadric. Thus the two fibers of the “canonical blowup” of W(0,3) over the points L and K-L, correspond to blowing down the fiber of the usual blowup along opposite rulings to obtain two different copies of P^1.

If C is a generic curve of genus 5, the subvariety W(1,4) is just the singular locus of W(0,4). The canonical blowup of W(0,4) is the Abel map C^(4) ≈ G(0,4)-->W(0,4), and the inverse image of W(1,4) is a P^1 bundle over it. In the usual blowup of W(0,4) along W(1,4), the inverse image of each point of W(1,4) is a smooth quadric surface. Thus the usual blowup of W(0,4) along W(1,4), is obtained from the canonical blowup G(0,4) by blowing up further along the inverse image C(1,4) of W(1,4). This situation is just the same as the previous one from genus 4, but with parameters. Equivalently, the canonical blowup of W(0,4) along W(1,4), is again obtained by first performing the usual blowup, and then partially blowing down the exceptional locus, a bundle of quadric surfaces over W(1,4), to obtain a P^1 bundle over W(1,4).

As Jack Huizenga suggested, in general perhaps one obtains the canonical blowup of W(r,d) by repeatedly blowing up in the usual sense along subvarieties W(s,d) with s > r, or their total transforms, and finally blowing down partially once again.