As I mentioned in my previous post, I am studying the article *Moduli of Enriques surfaces and Grothendieck-Riemann-Roch*.

The Grothendieck-Riemann-Roch theorem is applied there to show that, for any family of Enriques surfaces $f:Y\longrightarrow T$, the line bundle $$\mathcal{L} := R^0 f_\ast \Big ((\Omega^2_{Y/T})^{\otimes 2}\Big)$$ is a torsion line bundle, i.e., some tensor power $\mathcal{L}^{\otimes n}$ is isomorphic to the structure sheaf on $T$.

To this extent, one applies GRR to the morphism $f$ and the structure sheaf $\mathcal{O}_Y$. The problem I have now is with the "relative tangent sheaf". I am guessing this is the the quotient sheaf $$f^\ast \mathcal{T}_T/\mathcal{T}_Y.$$

**Q1**. Why is this well-defined? That is, why do we have an injection $ \mathcal{T}_Y \longrightarrow f^\ast \mathcal{T}_T$?

**Q2**. How can one determine the Chern classes of $\mathcal{T}_f$ by means of the fibres? That is, can one use the structure on the fibres (Enriques surfaces) to determine $c_i(\mathcal{T}_f)$?

**[New questions]**

**Q3**. Let $E$ be a fibre of $f:Y\longrightarrow T$ with injection $i:E\longrightarrow Y$. Is the ringmorphism $i^\ast:A^\cdot Y \longrightarrow A^\cdot E$ injective? If not, is it injective after tensoring with $\mathbf{Q}$?

Let $c_i=c_i(T_f)$.

**Q4**. We have two formulas from the GRR. The first is $1 = \frac{1}{12} f_\ast(c_1^2+c_2).$ This is the degree 0 part. The second comes from the degree 1 part and reads $0 = \frac{1}{24}f_\ast(c_1\cdot c_2).$ Now, why is $f_\ast(c_1^2) = 0$ as is suggested by the article?