Let $(G, \ast)$ be a group and let $H\le G$ and $K\le G$ be subgroups of $G$. Prove that $|HK|$=$\frac{|H|\cdot|K|}{|H\cap K|}$. Intuitively this is quite obviously true, as otherwise the products of all elements in the intersection of $H$ and $K$ would be counted twice, but no idea how to prove it! Any advice appreciated.

Let $V$ be an algebraic variety embedded in $\mathbb P^n$. Let $H$ be a hyperplane (that is to say, a variety defined by a single equation of degree $1$) such that for every $x \in H \cap V$, $x$ is a regular point of $V$, and $H$ does not contain the tangent space $T_V,x$. I […]

The question is as simple as that, but I have been trying to figure out an answer (and searching for it) with 0 results. I mean, given two triangles (in 2D) I want to find just a single point which they may have in common. Of course I have the long solution consisting of looking […]

In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$: \begin{equation} \mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g) \end{equation} where $P$ is the point in the intersection we are interested in, $\mathcal{O}_P$ is its local ring in $\mathbb{A}^2$ and $f$ and $g$ are the polynomials giving the two curves. On the […]

Let $S$ be an algebraic smooth surface over $\Bbb{C}$. Let $D\in\mathrm{Div}(S)$ be such that the complete linear system $|D|$ is base-point free and suppose $h^0(D)=N+1$ with $N>0$. To $D$ is then associated a morphism $\varphi:S\rightarrow\Bbb{P}^N$. By definition if $D$ is very ample then $\varphi$ is an embedding; in particular $\dim\varphi(S)=2$. Question: what are some possible […]

Consider the $k$-th chern class $c_k:=c_k(\mathcal{T}_{\mathbb{P}^n})$ of the tangent sheaf of projective space $\mathbb{P}^n=\mathbb{P}^n_\Bbbk$ over some (algebraically closed, if you want) field $\Bbbk$. I am then wondering what the degree of $\prod_{k=1}^n c_k^{\nu_k}$ is, given that $\sum_{k=1}^n k\nu_k=n$. For instance, I would already be happy to see how to compute $c_2c_1^{n-2}$. This is certainly well-known, […]

I need to do a polynomial interpolation of a set $N$ of experimental points; the functional form I have to use to interpolate is this: $$ f(x) = a + bx^2 + cx^4,$$ as you can see the coefficient that I need to find are just 3: $a, b, c$; however the points I have […]

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