I have a bunch of Differential Pulse Voltammetry data I am trying to import into Mathematica, and find the area between each curve and the blank. So far I have used the following commands:

"Data1 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-7-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data2 = Import[   "//Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-13-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data3 = Import[   "/Users/chelseaswank/Desktop/HSJ-6-26-19/HSJ-6-26-19-19-#2 - \ Sheet1.csv", "Table"] Data4 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-25-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data5 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-31-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data6 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-37-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data7 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-43-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data8 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-49-#2 - \ Sheet1 4.47.09 PM.csv", "Table"] Data9 = Import[   "/Users/chelseaswank/Desktop/untitled folder/HSJ-6-26-19-55-#2 - \ Sheet1 4.47.09 PM.csv", "Table"]"                 and                                                   "f1 = ListLinePlot[    Import["/Users/chelseaswank/Desktop/HSJ-6-26-19/HSJ-6-26-19-7-#2 - \ Sheet1.csv"]]; f2 = ListLinePlot[    Import["/Users/chelseaswank/Desktop/untitled \ folder/HSJ-6-26-19-13-#2 - Sheet1.csv"]]; Plot[{f1[x], f2[x], f1[x] - f2[x]}, {x, 0, 1.1},   Filling -> {{1 -> {2}}, 3 -> Axis}, PlotLegends -> "Expressions"]" 

## Counting elliptic curves by discriminant

Enumerating elliptic curves $$E/\mathbb{Q}$$ sorted by (the absolute value of) their minimal discriminants is a difficult open problem, as is the (likely easier) problem of counting elliptic curves $$E/\mathbb{Q}$$ given by a minimal Weierstrass model $$E_{A,B} : y^2 = x^3 + Ax + B$$, ordered by the model discriminant $$\Delta(E_{A,B}) = -16(4A^3 + 27B^2)$$.

For the question, it is not too hard to show that the number of curves $$E$$ with $$|\Delta(E_{A,B})| \leq X$$ is $$O(X)$$. Counting elliptic curves by their minimal Weierstrass models is essentially equivalent to counting monogenic cubic rings by discriminant, and the number of monic cubic rings of discriminant bounded by $$X$$ is surely less than the number of cubic rings having discriminant bounded by $$X$$, and the number of cubic rings having discriminant bounded by $$X$$ is $$O(X)$$ by the Davenport-Heilbronn theorem.

I am asking about whether there is any improvement over this bound: is it known that

$$\displaystyle N(X) = \#\{E_{A,B} : 16|4A^3 + 27B^2| \leq X\} = o(X)?$$

## Constructing elliptic curves defined over $\mathbb{Q}$ with fixed complex multiplication

I have the following problem: we know that for a field $$\kappa$$ of characteristic $$0$$ usually an elliptic curve $$E$$ defined over $$\kappa$$ is such that $$End(E)\cong \mathbb{Z}$$. This means that one cannot hope to find an elliptic curve with complex multiplication choosing it “randomly”.

Suppose that i want to produce and elliptic curve over $$\mathbb{Q}$$ whose $$End(E)$$ is and order in $$\mathbb{Q}(\sqrt{-D})$$: what are the methods currently known to do it? Is it possible to write explicitly the isogeny corresponding to $$\sqrt{-D}$$? (If it is in $$End(E)$$ and $$D$$ is not so large).

## The underlying curve of a family of genus zero $n$ punctured curves

Let $$X$$ be a curve of genus zero over an algebraically closed field $$k$$ so that $$X \cong \mathbb{P}_k^1$$. Let $$(C, s_1, \cdots, s_n)$$ a $$n$$ punctured genus zero curve over $$k$$ where $$s_i: k \to C$$ are sections. Sine $$C$$ is a genus zero over $$k$$, we have that $$C \cong \mathbb{P}_k^1$$.

Now let $$B$$ be an affine scheme over $$k$$ and suppose consider the family of $$n$$-punctured genus zero curves $$(C \to B, s_1, \cdots, s_n)$$ where $$s_i:B \to C$$. Let us always assume that $$n \ge 4$$.

At this point, I tend to come across statements inferring that since the fibers of the family $$(C \to B, s_i)$$ have non non-trivial automorphism $$C \cong \mathbb{P}_B^1$$. Here I by no means am claiming that the family $$(C \to B, s_i)$$ of $$n$$-punctured curves is trivial but rather that the (total space) of the family of genus zero curves over $$B$$ “underlying” the family $$(C \to B, s_i)$$ is trivial.

Why does the automorphism group of the fibers of a family $$(C \to B, s_i)$$ being trivial imply that the underlying family of genus zero curves over $$B$$ is trivial? Again I mean this in the sense that the total space $$C \cong \mathbb{P}_B^1$$.

## Singularity of Brill-Noether sub varieties of Picard varieties of smooth curves

Suppose $$C$$ is a smooth projective curve over complex numbers. The singularities of the theta divisor $$\Theta$$ in $$Pic^{g-1}(C)$$ is described in the literature. It is $$W^{1}_{g-1}=\{l\in Pic^{g-1}(C): h^0(l)\geq 2\}$$. I could find in the literature the expected dimension results. Are the singularities of $$W^1_{g-1}$$ known.

Question: What is the (exp)dimension of $$Sing(W^1_{g-1})$$, for a generic curve ?

## Do negative indecomposable bundles on curves have sections?

Let $$X$$ be a smooth projective curve, and $$E$$ an indecomposable vector bundle on $$X$$ with $$\mathrm{deg} E<0$$. Is it true that $$H^0(X,E)=0$$?

This is true if $$E$$ is a line bundle, which means it is also true whenever $$X$$ is $$\mathbb{P}^1$$, since all vector bundles split here.

It is also true by results of Atiyah if $$X$$ is an elliptic curve. What about for curves of higher genus?

The assumption that $$E$$ is indecomposable is of course necessary.

## Reference request: smooth affine curves are planar

Let $$X\rightarrow\mathrm{Spec}\:\mathbb{C}$$ be an affine smooth morphism of relative dimension$$\leq 1$$. What is a reference for the fact that there exists a $$\mathbb{C}$$-locally closed immersion $$X\rightarrow \mathbb{P}^2$$?

## Singularities of curves that are moving

Let $$k$$ be an algebraically closed field, let $$d\ge 2$$ be an integer and let $$f,g\in k[x,y,z]$$ be two homogeneous polynomials of degree $$d$$ without common factor.

We want to know what are the singularities of the curve $$C_{[\lambda:\mu]}$$ given by $$\lambda f+\mu g=0$$, for a general $$[\lambda:\mu]\in \mathbb{P}^1$$. If a point $$p\in\mathbb{P}^2$$ is a singular point of the curves $$C_{[1:0]}$$ and $$C_{[0:1]}$$ given by $$f=0$$ and $$g=0$$ then it is of course singular for each $$C_{[\lambda:\mu]}$$. If $$\mathrm{char}(k)=0$$, then by Bertini there are no other singularities. If $$\mathrm{char}(k)=p>0$$, it is false: take for instance $$f=x^p$$ and $$g=y^p$$. Are all counterexamples of this type ? One can of course replace $$p$$ by a power of $$p$$ and maybe do some more general examples. For instance, if $$d<2p$$, is the case $$f=x^p$$ and $$g=y^p$$ the only possibility (up to change of coordinates)?

## Picard group of the moduli space of semistable rank 2 parabolic vector bundles over smooth complex projective curves with trivial determinant

I am looking for the Picard group of the moduli space of semistable rank 2 parabolic vector bundles over smooth complex projective curves with trivial determinant.

Having determinant trivial, I believe we can consider the bundles as parabolic $$SL(n,\mathbb{C})$$-bundles as well. The closest result have found in this regard is a paper by Laszlo and Sorger, ‘The Picard group of the moduli of G-bundles on a curve’, but they only talk about quasi-parabolic bundles rather than parabolic bundles.

My question is: Is the picard group known already? Does anyone know of any reference? I need it urgently if possible. Any help would be appreciable.

## All curves over an infinite field embed into the projective space

Let $$k$$ be an infinite field. Let $$X$$ be a separated scheme of finite type over $$k$$. Assume $$X$$ have relative dimension $$\leq 1$$. Does there exist a locally closed immersion $$X\rightarrow \mathbb{P}^3_k$$? Is there a published reference containing a complete proof?

I think over finite fields you can have too many points so this can fail.

EDIT: actually, I think if you have a very singular curve the dimension of the tangent space can be problematic. What are the minimal smoothness assumptions on $$X$$ under which this is true?