By Daniel Huybrechts, we have:
Definition: Let $E$ be a coherent sheaf of dimension $d = \text{dim}X$. The degree of $E$ is defined by: $$\text{deg}(E) = \alpha_{d-1}(E) – \text{rk}(E).\alpha_{d-1}(\mathcal{O}_{X})$$
where $\alpha_{i}$ are the coefficients of the Hilbert polynomial: $$P(E) = \displaystyle \sum_{i=0}^{\text{dim}X}\alpha_{i}(E)\dfrac{m^{i}}{i !}$$
Let $F$ be a coherent sheaf in $X$ such that $\text{dim}F \neq \text{dim}X$.
First question: In this case, what is the definition of $\text{deg}F$? Is this degree defined with respect to a fixed line bundle?
Let $X$ be a smooth projective scheme and $Y$ a smooth projective subscheme. Consider the blow-up $$\pi : \widetilde{X} \longrightarrow X$$ of $X$ along of $Y$ with exceptional divisor $E$.
Consider the line bundle $\widetilde{L} = \pi^{*}\mathcal{O}_{X}(k) \otimes \mathcal{O}_{\widetilde{X}}(-E)$ in $\widetilde{X}$.
Second question: If the second part of the first question is true, then what would be the degree of a coherent sheaf $\widetilde{F}$ in $\widetilde{X}$ with respect to the line bundle $\widetilde{L}$?
Suggestions and references on this subject will be welcome.
Thank you very much.
Best Answer
There are two things to keep in mind. First, we always need to fix an ample line bundle to speak about stability. Secondly, Intersection Theory is made a way that you can work (at least if $X$ is smooth) on the Grothendieck group $K_0(X)$, so one might consider locally free resolution.
Fix an ample line bundle $H$. To define the degree of any coherent sheaf $F$, you can consider a locally free resolution $$0\to F_m \to \cdots \to F_1 \to F_0 \to F$$ and define $\det(F)=\prod \det(F_i)^{(-1)^i}$. Then you define $$\deg(F) = c_1(\det(F))\cdot H^{\dim(X)-1}$$ This definition agrees with the one you state in the case of maximal dimension (see http://www.math.harvard.edu/~yifei/tensor_char_zero.pdf Lemma $1.20$).
Example: If you consider an effective divisor $D\in Pic(X)$, you have the exact sequence $0 \to \mathcal{O}_X(-D) \to \mathcal{O}_X \to \mathcal{O}_D \to 0$, thus $c_1(\mathcal{O}_D)=D$ (as a cycle) and thus $\deg_H(\mathcal{O}_D)=D\cdot H^{\dim X -1} =\deg_H(D)$ (counting multiplicities and taking the sum on irreducible components).
_**Example in a blow-up:*__ Now consider the simple example of $\widetilde{X}=Bl_{p}\mathbb{P}^2 \to \mathbb{P}^2$, the blowup of $\mathbb{P}^2$ at a point $p$. If I take a the strict transform $\widetilde{C}$ of an effective divisor $C\in Pic(\mathbb{P}^2)$, it is still an effective divisor, and $\widetilde{C}=\pi^*C-rE$ where $r$ is the multiplicity of $C$ at $p$. Then there is a resolution $$0 \to O_{\widetilde{X}}(-D) \to O_{\widetilde{X}} \to O_D \to 0$$ and we obtain once again $c_1(\mathcal{O}_{\widetilde{C}})=\widetilde{C}$. Thus if you fix your ample divisor $T=m\pi^*H - E$ on $\widetilde{X}$, you get
\begin{eqnarray} \deg_T(\mathcal{O}_{\widetilde{C}}) &=& \widetilde{C}\cdot T\\ &=& (\pi^*C-rE)\cdot(m\pi^*H-E)\\ &=&m\deg_H(C)-r. \end{eqnarray}
Example for vector bundle : Consider $\pi : \widetilde{X} \to X$ the blow-up of a smooth projective scheme along a smooth projective subscheme and write $E$ the exeptional divisor. Fix an ample divisor $H$ on $\widetilde{X}$. For a vector bundle $V$ of rank $r$ on $X$, choose a resolution $$0 \to V_n \to \cdots \to V_0 \to V \to 0,$$ thus $\deg_H(V)= c_1\left(\prod (\det V_i)^{(-1)^{i}}\right)\cdot H^{\dim X-1} = \sum (-1)^i c_1(\det V_i)\cdot H^{\dim X-1}$.
Now define $\widetilde{V}=\pi^* V \otimes O_{\widetilde{X}}(-lE)$ for some $l\in\mathbb{Z}$. As pullback is exact when apply to locally free sheaves, you have the resolution $$0 \to \pi^* V_n \to \cdots \to \pi^* V_0 \to \pi^* V \to 0.$$ Fix the ample line bundle $T=m\pi^* H - E$ on $\widetilde{X}$. As $c_1(\widetilde{V})=c_1(\pi^*(V)\otimes \mathcal{O}_{\widetilde{X}}(-lE) = c_1(\pi^*V)-rlE$, we obtain
\begin{eqnarray} \deg_T (\widetilde{V}) &=& (c_1(\pi^* V)-rlE)\cdot T^{\dim X -1}\\ &=&\sum (-1)^ic_1( \det \pi^* V_i)(m\pi^* H^{\dim X-1})+rlE^2 \\ &=& m\deg_H(V)+rlE^2 \end{eqnarray}
In more general settings, I don't know if things go so well. You need to find a locally free resolution of the strict transform, which might be harder as pullback is not exact in general.