Articles of intersection theory

Definition of intersection multiplicity in Hartshorne VS Fulton for plane curves

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 […]

Dimension of the image of the morphism associated to a Divisor

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 […]

Chern numbers of Projective Space

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, […]

Trouble doing polynomial interpolation

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 […]