Articles of algebraic curves

cohomology of the pullback of a sheaf

Let $X$ a degree $d$ curve in $\mathbb{P}^n$ not contained in any hyperplane and $i:X\hookrightarrow \mathbb{P}^n$ the corresponding closed immersion. Then I would like to prove that $dim H^0(X, i^{*}O_{\mathbb{P}^n}(1))=n+1$. I think that the proof of this fact lies on the fact that it is not contained in any hyperplane so you can count linearly […]

Why is a genus 1 curve smooth and is it still true for a non-zero genus one in general?

In the very commonly used J. Silverman’s AEC an elliptic curve is defined as a genus 1 projective curve with a fixed point 0. In all the other books I looked at it is defined to be (also) smooth. By the way in AEC it is given a proof of the fact that a genus […]

$V=\mathcal{Z}(xy-z) \cong \mathbb{A}^2$.

This question is typically seen in the beginning of a commutative algebra course or algebraic geometry course. Let $V = \mathcal{Z}(xy-z) \subset \mathbb{A}^3$. Here $\mathcal{Z}$ is the zero locus. Prove that $V$ is isomorphic to $\mathbb{A}^2$ as algebraic sets and provide an explicit isomorphism $\phi$ and associated $k$-algebra isomorphism $\tilde{\phi}$ from $k[V]$ to $k[\mathbb{A}^2]$ along […]

Solving the curve equation for logarithmic decay using two anchor points.

I would like to have an adaptable logarithmic curve equation that I can then find y for any value of x. I have two points (x1,y1) and (x2,y2). My data requires constant decay (financial discounting of cash flows) and crosses the y axis. So with the two anchor points, what would be the generalised line […]

what is genus of complete intersection for: $F_1 = x_0 x_3 – x_1 x_2 , F_2 = x_0^2 + x_1^2 + x_2^2 + x_3^2$

In the below problem I want to answer second part: Show that the curve in $\mathbb{P}^3$ defined by the two equations $x_0 x_3 = x_1 x_2$ and $x_0^2 + x_1^2 + x_2^2 + x_3^2 = 0$ is a smooth complete intersection curve. What is its topological genus? but I don’t know how can I calculate […]

Pole set of rational function on $V(WZ-XY)$

Let $V = V(WZ – XY)\subset \mathbb{A}(k)^4$ (k is algebraically closed). This is an irreducible algebraic set so the coordinate ring is an integral domain which allows us to form a field of fractions, $k(V).$ Let $\overline{W}, \overline{X}$ denote the image of $W$ and $X$ in the coordinate ring. Let $f=\dfrac{\overline{W}}{\overline{X}}\in k(V).$ I want to […]

Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn’t find it in Diamond and Shurman, and neither could I find an explicit formula with a simple google search. Certainly, there should be some explicit expression, no?

Why $y=e^x$ is not an algebraic curve?

Why $y=e^x$ is not an algebraic curve over $\mathbb R$? I can say that is not a algebraic curve over $\mathbb C$ because $e^x$ is a periodic function, but what about $\mathbb R$? EDIT: I don’t want to use trascendence of $e$. Or, I can ask this question for $y=2^x$. UPDATE: Can we just say […]

projective cubic

I have some difficulties to prove that the image of the function $f:\,\mathbb P^1\longrightarrow\mathbb P^3$ such that $$(u,v)\longmapsto (u^3,\,u^2v,\,uv^2,\,v^3)$$ is the algebraic projective set $$V(XT-YZ,\, Y^2-XZ,\,Z^2-YT)$$ Clearly I have problem to prove that the algebraic projective set is contained in the image of $f$. In particular “solving brutally” the polynomial system I’m losing my mind […]

Normal bundle of twisted cubic.

Let $C$ be a twisted cubic in $\mathbb P^3$. I’d like to compute the splitting type of normal bundle $N_{C/\mathbb P^3}$? I understood that $T_{\mathbb P^3}|_C=\mathcal O(4)^{\oplus 3}.$ So we have an short exact sequence $$ 0 \to \mathcal O(2) \to \mathcal O(4)^{\oplus 3} \to N_{C/\mathbb P^3} \to 0. $$ So $N_{C/\mathbb P^3} =\mathcal O(4) […]