“Canonical blowup” of W(r,d), versus usual blowup.
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.
Note also, the usual blowup of W(r,d) at a point L, depends only on the intrinsic geometry, whereas the “canonical blowup” depends on the representation of W(r,d) as a degeneracy locus for maps of sections of line bundles on C. The intrinsic nature of this representation depends in some sense on the Torelli theorem which asserts that the polarized Jacobian variety of C has only one such representation, i.e. it is not isomorphic to the polarized Jacobian variety of some other curve.
W(r,d) is by definition that subset of the Picard variety of a curve C, consisting of points corresponding to line bundles L of degree d with at least r+1 independent sections. The goal is to study the relation between the intrinsic geometry of this set and the analytical properties of L. In the classical case r=0, if g is the genus of C, Riemann showed the multiplicity of W(0,g-1) at L is h^0(C,L), and Kempf generalized work of Mumford and Andreotti-Mayer to compute the projective tangent cone as the union of the spans of the divisors of sections of L, in the “canonical space” PH^0(C;K)* of the curve C. He also did the case of W(0,d).
Since G(0,d)-->W(0,d) is a resolution with smooth fibers, the induced map from the usual blowup of G(0,d) along |L| to the usual blowup of W(0,d) at L is computable as the map induced by the derivative of the Abel map, from the projective normal bundle to the fiber |L| in G(0,d), to the projective tangent cone of W(0,d) at L, i.e. to the fiber of the usual blowup of W(0,d) at L. Thus the usual blowup of W(0,d) is computed as the image of the known normal bundle along |L| in the “canonical blowup” G(0,d).
This method of computing the usual blowup of W(r,d) is useful only when one can explicitly compute the usual blowup of the canonical blowup G(r,d) along the fiber over L, e.g., when the canonical blowup G(r,d) is smooth along the fiber. Thus on page 240 of ACGH, in Theorem VI.2.1, they generalize Kempf’s theorem to compute the projective tangent cone to W(r,d), i.e. the fiber of the usual blowup, in that case only. (Notice when r = 0, then G(0,d) ≈ C^(d), the symmetric product of C, which is smooth, so the method always applies in the case considered by Kempf.) Their method of proof, using their Lemma II.1.3, assumes birationality of the canonical parametrizing map. This explains their definition of G(r,d), rather than restricting the usual Abel map to C(r,d) which only gives a generic P^r fibration.
The simplest example is the model for the canonical blowup of a general double point on the theta divisor W(0,g-1) of a non hyprelliptic curve, namely (a pullback of) the kernel resolution of the discriminant locus {xy-zw = 0} in Mat(2x2). This “canonical blowup” inserts the projectivized kernel of the zero matrix, a copy of P^1, over the singular point at the origin. The induced map of usual blowups of these two varieties, maps the normal bundle along this P^1, isomorphic to P^1xP^1, onto the projectivized tangent cone at the origin of the discriminant locus. In this case the induced map of usual blowups is an isomorphism, i.e. the usual blowup of {xy-zw=0} at the origin is isomorphic to the blowup along the exceptional P^1 of the “canonical blowup”. (In general it seems only that there is a map parametrizing the usual blowup, by the usual blowup of the canonical one.)
The previous example is the precise local model for the Abel resolution G(0,3)-->W(0,3) of the theta divisor for a non hyperelliptic curve C of genus 4 by the symmetric cube C^(3) ≈ G(0,3), of the curve C. I.e. the Abel resolution C^(3) contains the fiber |L| ≈ P^1 over a line bundle L with h^0(L) = 2, and is isomorphic to the “canonical blowup”. The usual blowup of the theta divisor W(0,3) inserts a copy of P^1xP^1.
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.
In the case of the theta divisor of a Prym variety there is no Torelli theorem, and Brill Noether representations and the associated Abel-Prym maps are neither intrinsic to the geometry of the polarized Prym variety, nor birational. Indeed since the Prym theta divisor lies inside W(1,g-1), the Abel-Prym map is generically a P^1 fibration. For these reasons in Appendix C of chapter VI of ACGH there were no analogous results given for Prym varieties. More recently some intrinsic information about the Prym theta divisor, such as the degree of the tangent cone at a point, has been computed in terms of invariants of the curves associated to any representation.
http://annals.math.princeton.edu/2009/170-1/p05
The primary obstacle to parametrizing the tangent cone is the source spaces of the Abel-Prym map, the Prym divisor varieties, are not always smooth. When a smooth one exists one obtains an analogous parametrization of the tangent cone of the Prym theta divisor, i.e. of the fiber of the usual blowup, by the linear normal bundle to the fiber of the Abel-Prym map, although not a birational one. Even in some non smooth cases, such as a Prym representation of the intermediate Jacobian of a cubic threefold, one can still calculate the parametrization of the tangent cone to the Prym theta divisor by the non linear normal cone along the fiber of the Abel-Prym map.
