ag.algebraic geometry – Do singular fibers determine the elliptic K3 surface, generically?

Elliptic K3 surfaces. Let $X$ be a general projective elliptic K3 of Picard rank two. Assume that singular fibers of the elliptic fibrations are of type $A_1$ so that by the Euler characteristic count there are $e(X) = 24$ singular fibers. I do not require existence of a section in the definition of an elliptic K3 surface.

Question: do these $24$ points determine the general such elliptic K3 uniquely, or at least up to finitely many choices? If not, what can be said about the dimension of the moduli space of those elliptic K3 surfaces with a prescribed set of $24$ singular fibers?

Any suggestions or references welcome.

linear algebra – Connection of the singular value before and after normalization

Given a matrix $P = mathbb{R}^{n times d}$, we can get $P = U Sigma V^T$ by using SVD.

Let’s say, we have another matrix $P’ = mathbb{R}^{n times d}$, it is the $P$ matrix with normalization along the second dimension. So each row of $P$ is a unit vector. We can also do SVD with $P’ = U’ Sigma’ V’^T$.

Is there any connection between the singular value $Sigma$ and $Sigma’$?

dg.differential geometry – Characterization of planar domains onto which a unit disk can be mapped with constant singular values

It can be shown that there are (smoothly bounded, Jordan) domains $Esubset mathbb{R}^2$ which are $textit{not}$ images of mappings $f$ from the unit disk (or any other planar domain), such that $mathrm{d}f$ has fixed, constant singular values. That is, such mappings have ‘limited transformation capability’.

On the other hand, I wonder if a characterization of all those domains onto which the unit disk (or more generally, any planar domain) can be transformed to by such $f$‘s.

differential equations – Local Series solution at singular point for system of first order ODEs

I want to calculate Psi(z) in the equation

D(Psi(z), z) + A(z).Psi(z) == psi(z)

around a given point p, where A(z) is an $ntimes n$–matrix, psi(z) is an $n$–vector and hence Psi(z) also needs to be an $n$–vector. The matrix A(z) and vector phi(z) can both be singular at p, but neither has an essential singularity. Can you please help me solve this problem efficiently?

The goal is then to take Psi(z) and multiply it with some given phi(z), and then take the residue at p. Therefore ideally the outcome is a SeriesData object around p, or something similar to that. Note that p generally is a point on $CP^1$, so p can be ComplexInfinity.

I have tried a few things and some were succesfull in subcases, but while those often worked in reasonable time for $n=1$ they took unreasonably long for $n>4$ or had similar issues.

  • I generally wrote A(z) as a list of SeriesData objects with coefficients Psico(l, k) $(l in {1, dots, n})$ and expanded A(z) and psi(z) using Series and took the list of their coefficients Acoef and psiCoef, which reduces the equation to something of the form
(a+1) Psico(j, a+1) + Sum(Acoef(b)((j, k)) * Psico(k, a-b), {b, low, high}, {k, 1, n}) == psiCoef(j, a).
  • For A(z) non-singular in z, the equation becomes an easy recursive equation in the expansion coefficients of Psi(z). This can be solved either using recursive methods (such as RecurrenceTable or a For loop by hand) or by writing the equations as a triangular matrix and solving that (using e.g. LinearSolve). However, this ran into problems when A(z) is singular at p as in that case the matrix is no longer triangular and equivalently the recursive method requires coefficients which we do not yet know.
  • For $n$ small, the matrix could still forcefully be solved using LinearSolve, but for large $n$ that becomes incredibly slow. Perhaps that is because my code was too naive, and there are some tricks to convince LinearSolve that the matrix is nearly triangular?
  • I then tried using methods such as AsymptoticDSolveValue, but that persistently tried to approximate Psi(z) with a polynomial (not using singular terms). I am using Mathematica 12.0 Student Edition (where AsymptoticDSolveValue is Experimental); I have understood this method has been changed slightly in the newer version so perhaps this is now better but I have no acces to newer version of Mathematica.

Finally remark that there is of course not just a single solution, but your can generally add the solution to the homogeneous equation. This can be ignored, as it is mathematically expected that the homogeneous solution does not contribute to the later residue. Therefore not all solutions are needed, but just one solution should be sufficient. Of course, the homogeneous solution generally has an essential singularity, so that solution can typically be avoided somewhat by choosing some sufficiently low coefficient Psico(k, low) and fixing it to be 0 by hand.

Could you please help me? Thank you very much in advance!

A minimal example of A(z), psi(z) and p could be: (x is an external constant)

n=3;
A(z_) = {{z, 1/z, 1/(z - 1)}, {z^2, x z, (1 - x) z}, {0, 1/(x z), z}};
psi(z_) = {1, 1/z, 1/(z-1)};
p=1;

Boundary conditions for singular Sturm-Liouville problem (boundary behavior of eigenfunctions)

I am not at all an expert in Sturm-Liouville theory, but I ended up on the following Singular Sturm Liouville problem:
begin{equation}label{1}
(1) y”(t)+frac{theta'(t)}{theta(t)}y'(t)+lambda y(t)=0,, tin(0,R)
end{equation}

which can be rewritten as
$$
(theta(t) y'(t) )’+lambdatheta(t)y(t) =0,, tin(0,R).
$$

The weight function is strictly positive and smooth on $(0,R)$, and vanishes at $t=0$ and $t=R$, which are then singular endpoints. I don’t have much information on $theta$ except that it has polynomial behavior near the endpoints, namely
begin{equation}label{2}
(2) theta(t)sim_{trightarrow 0^+} t^a {rm and} theta(t)sim_{trightarrow R^-} (R-t)^a.
end{equation}

The crucial point is that I need the boundary conditions (which I can’t impose in principle)
$$
y'(0)=y'(R)=0
$$

In specific cases I know how to handle this (e.g., when $theta(r)=sin(rpi/R)^a$ by suitable change of variables I end up on a Jacobi equation).

In the general case, from Sturm-Liouville theory (if I am not wrong, otherwise please correct me), if $1leq a<3$, the endpoints are of LC type, while they are LP type if $ageq 3$.

In the LC case, the natural choice for boundary conditions for me is $theta(t)y'(t)rightarrow 0$ as $trightarrow 0^+$ and $R^-$.

In the LP case such boundary conditions are in a sense authomatic and no other options are available. In both cases I have discrete spectrum, made of simple non-negative eigenvalues $lambda_n$ diverging to $+infty$ (due to the behavior of $theta$ near the endpoints I can see that in the LP case the essential spectrum is empty).

The only point that is left to prove is: do the eigenfunctions $y_n$ satisfy also $y_n'(0)=y_n'(R)=0$ (and not only $theta(t)y_n'(r)rightarrow 0$ as $trightarrow 0^+$ and $R^-$)?

An example to motivate this (perhaps experts have already the answer): in the simple case
$$
y”(t)+frac{a}{t}y'(t)+lambda y(t)=0,, tin(0,1)
$$

where $t=0$ is the only singular endpoint, I can put the boundary condition $t^ay'(t)rightarrow 0$ as $trightarrow 0$ (and any “classical” BC at t=1) if $1leq a<3$. The same b.c. at $t=0$ is authomatic if $ageq 3$. Then I have in both cases that $y_n(t)=x^{frac{1-a}{2}}J_{frac{a-1}{2}}(sqrt{lambda_n}x)$, where $lambda_n$ are provided by imposing the classical b.c. at $t=1$ (for $lambda_0=0$, $y_0=1$).

These eigenfunctions satisfy the condition $t^ay_n'(t)rightarrow 0$ (which I have imposed in the case $1leq a<3$, and which is authomatic for $ageq 3$). Not only, they satisfy $y_n'(0)=0$ (but this I can recover from well-known theory of Bessel functions, not from the form of the problem).

To summarize, my question is the following:

“Do the eigenfunctions $y_n$ of (1) with weights satisfying (2), and with $theta(t)y_n'(t)rightarrow 0$ as $trightarrow 0^+$ and $R^-$ satisfy also the condition $y_n'(0)=y_n'(R)=0$?”

This seems quite reasonable and intuitive to me. I would appreciate some reference where to find this kind of result.

ag.algebraic geometry – Cycle class map for singular varieties

I am reading the cycle class map for singular projective varieties as mentioned by Laterveer in this article (see Definition $1$). The article does not define the map but refers to an article of Totaro, which also does not define the map but refers to the article “Non-archimedean Arakelov theory” by Bloch, Gillet and Soule, which is published in JAG in 1995 and is not available online. Can someone give a bit more details on this map (or an alternate reference)? For example, the authors say that this map is functorial. I do not quite understand what this means, as the cohomology group is functorial under pull-back and the Chow-group is functorial under proper pushforward and flat pull-back? Moreover, does the image of the composition
$$mathrm{A}^p(X) xrightarrow{cl} mathrm{Gr}^{W}_{2p}H^{2p}(X,mathbb{Q}) xrightarrow{rest.} mathrm{Gr}^{W}_{2p}H^{2p}(X_{mathrm{smooth}},mathbb{Q})$$
is the same the usual cycle class map
$$A^p(X_{mathrm{smooth}}) xrightarrow{cl} H_{2n-2p}^{mathrm{BM}}(X_{mathrm{smooth}}) xrightarrow{P.D.}
H^{2p}(X_{mathrm{smooth}})?$$

What is the pull-back of the image of the singular cycle class map to the cohomology on
the resolution of singularities?

ag.algebraic geometry – Poisson rank on singular schemes

My question is related to the Poisson structure on singular schemes. I am reading the paper ” Poisson Modules and degeneracy loci” by Marco Gualtieri and Brent Pym. On page 633 they define the Degeneracy loci of a Poisson structure.

Let $(X, sigma)$ be a Poisson scheme. The degeneracy loci $D_{2k}(sigma)$ of $sigma$ is the locus where the morphism $sigma^{#}: Omega^1_Xrightarrow T_X:=Der(mathcal O_X, mathcal O_X)$ has rank at most $2k$.

My question: What is the definition of rank of such a morphism?

Because the sheaves $Omega^1_X$ and $T_X$ may not be locally free. Moreover, $Omega^1_X$ may not be even torsion-free. It is not clear to me what is the definition of the rank in this case.

I understand that there are other definitions of Degeneracy loci. But I am curious to know the definition of the rank here. Thanks in advance.

pr.probability – Classifying non atomic singular measures up to topological conjugacy

Write $mathcal S$ for the set of probability measures on $[0, 1]$ that are non atomic and singular with respect to Lebesgue measure.

Two measures $mu$ and $nu$ in $mathcal S$ are said to be topologically conjugate if, denoting by $F_mu$ and $F_nu$ their respective distribution functions, there exist homeomorphisms $h$ and $g$ of $[0, 1]$ such that $h F_mu = F_nu
g$
.

Topological conjugacy of measures thus forms an equivalence relationship on $mathcal S$. Intuitively, two measures are topologically conjugate if they differ only by a continuous change of coordinates in the domain and a continuous change of measure values.

Can we classify the measures in $mathcal S$ up to topological conjugacy?

ag.algebraic geometry – Singular del Pezzo surfaces and degeneration of root systems

Let $S$ be a smooth del Pezzo surface of degree $d$ and $K_S^*$ the anticanonical class. It is well known that the set of classes

$$R(S)={alphain H^2(S,mathbb Z)|alpha^2=-2,alphacdot K_S^*=0},$$

together with intersection pairing is isomorphic to a root system $R_d$ of rank $9-d$, where $R_d$ from $d=1$ to $6$ is $E_8,E_7,E_6,D_5,A_4,A_2times A_1$. Moreover, Each root can be written as $(L_1)-(L_2)$, where $L_1,L_2$ are two disjoint lines on $S$.

I’m particularly interested in the $d=3$ case, where $S$ is a cubic surface, and the corresponding root system $E_6$ has 72 roots. I’d like to know

Question 1: Is there a notion of the root system on singular del Pezzo surfaces? Is there any related reference?

Now let $S$ be a normal del Pezzo surface of degree $d$ with at worst ADE singularities. Then there is a minimal resolution $pi:S’to S$, where $S’$ is a weak del Pezzo surface, which arises blowing up $9-d$ bubble points on $mathbb P^2$. Moreover, $pi$ is given by anticanonical map $|K_{S’}|$ and contracts all effective $(-2)$ curves. Since $S’$ is deformation equivalent to a smooth del Pezzo surface, the root system of $S’$ is also isomorphic to $R_d$. According to Dolgachev’s book, section 8.4 and 9.2.3, and also see Martin Bright’s answer, the set of effective $(-2)$ curves on $S’$ correspond to a sub root system $R_esubseteq R_d$. Such sub-root systems are classified. Moreover, the rank $r=text{rank}(R_e)$ coincides with the number of the $(-2)$-effective curves.

To me, the roots on singular $S$ should be a degeneration of the root system $R_d$. It should be certain “complement” of the sub-root system $R_e$ of the $(-2)$ curves occur in the resolution. However, the orthogonal complement does not seems to the correct notion, since strict transform of a line through the singular point will have nontrivial intersection with the exceptional curves, and there should be “roots” as differences of lines through the singularity.

The notion of root system $R(S)$ on singular $S$ that I have in mind is a quotient:

Definition 1: Let $R(S)$ be the quotient set $R_d/R_e:=R_d/sim$, where the equivalence relation on $R_d$ is $r_1sim r_2$ iff $r_1-r_2in R_e$.

Such a quotient is just a set and does not have a root system structure in general. (Although the quotient seems to behave well for type $A$ root systems, so for $dge 5$ cases, the quotients correspond to the Dynkin diagram by removing the sub-Dynkin diagram corresponding to $R_e$).

For example, for a cubic surface with an $A_1$ singularity, $R(S)=E_6/A_1$ has 50 “roots”; for a quartic del Pezzo surface with an $A_1$ singularity, $R(S)=D_5/A_1$ has 26 “roots”. As far as I know, 50 and 26 are not the number of roots of any root systems (even reducible).

The reason for me to consider this quotient is that the class group $text{Cl}(S)$ arises as a surjection $text{Pic}(S’)twoheadrightarrow text{Cl}(S)$ with kernel generated by the set of effective $(-2)$ curves. However, I haven’t seen any reference ever discussed such quotient.

Question 2: Is there a Lie theory interpretation of the quotient in Definition 1?

Any suggestions and comments are welcome!