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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.

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1 Answer 1

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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.

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  • $\begingroup$ I am interested in the variety $\widetilde{X}$. According to your answer above, what would then be $\text{deg}(\widetilde{L})$? $\text{deg}(\widetilde{L}) < 0$? $\text{deg}(\widetilde{L}) = 0$? Or $\text{deg}(\widetilde{L}) > 0$? (where $\widetilde{L}$ is the line bundle as in the original question.) Thank you very much. $\endgroup$ Oct 1, 2019 at 10:55
  • $\begingroup$ For example. If $X = \mathbb{P}^{n}$ and $H$ is an ample divisor on $X$, the divisor $D = m\pi^{*}H - E$ is very ample on $\widetilde{X}$ for all integers $m$ sufficiently large, according Hartshorne. Consider the line bundle associated $\mathcal{L} = \mathcal{O}_{\widetilde{X}}(D)$. In this case what would it be $\text{deg}(\mathcal{L})?$. $\endgroup$ Oct 1, 2019 at 11:02
  • $\begingroup$ Could you please display an example in $\widetilde{X} $ as you displayed it for $ X$? I have doubts if a bidegree appears due to the exceptional divisor, so I ask you, please, an example of how to explicitly calculate the degree in $\widetilde{X} $. Thank you very much. $\endgroup$ Oct 1, 2019 at 11:08
  • $\begingroup$ I edited my answer to add an easy example. I don't really know how you can do in full generality, but there are probably results about it. $\endgroup$ Oct 1, 2019 at 14:41
  • $\begingroup$ Really, I wonder if this works for higher dimensions. For example, Let $C \subset X = \mathbb{P}^{3}$ be a smooth algebraic curve and $\pi: \widetilde{X} \longrightarrow X$ the blowup along of $C$. In this case, $C$ it is not a divisor. Do you have an idea how we can apply your answer in this case? $\endgroup$ Oct 1, 2019 at 23:10

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