Finding area between curves -loading csv data

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"]" 

and I get an axis and a green line. Please help.

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).

Thank you for your time.

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.

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?