# Is this $\mathbb P^1$-fibration over $\mathbb P^1$ a flat family?

This example is a test of several quantities invariant in a flat family.

Let $$X$$ be a projective variety, let $$D$$ be a divisor on $$X$$ with $$\dim|D|\ge 1$$. Choose a pencil $$\{X_t\}_{t\in\mathbb P^1}$$ inside $$|D|$$. This is in particular an algebraical family, therefore a flat family. We know Hilbert polynomial is invariant in a flat family, therefore all quantities arising from Hilbert polynomial, including dimension, degree and arithmetic genus, should be invariant in a such a family.

Here is an example seems contradicting to me: Let $$p:X\to \mathbb P^1\times \mathbb P^1$$ be the surface blowing up at a point, and let $$\pi:X\to \mathbb P^1$$ be the composite $$p_1\circ p$$ with $$p_1$$ the projection to the first coordinate. ($$X$$ is indeed a Hirzebruch surface $$\Sigma_1$$, but we are considering its $$\mathbb P^1$$-fibration in a different way.)

Then the map $$\pi$$ gives us a pencil family of divisors on $$X$$, with general fiber $$X_t=\mathbb P^1$$, and special fiber $$X_0=\mathbb P^1\cup \mathbb P^1$$, two $$\mathbb P^1$$ intersecting at a point. Now, let's check the those quantities mentioned above.

Obviously, the dimension is always 1 which doesn't change. The arithmetic genus is also the same by the answer of this post. However, the degree of $$X_t$$ is $$1$$, while the degree of $$X_0$$ is $$2$$ (The degree is additive on irreducible components, I believe.)

So my question is:

(1) Does $$\pi: X\to \mathbb P^1$$ gives a flat family of divisors?

(2) Is it true that $$\deg(X_0)=2$$?

The degree is computed with respect to some (ample) line bundle. For instance, if $$h_1$$ and $$h_2$$ are the hyperplane classes of factors of $$\mathbb{P}^1 \times \mathbb{P}^1$$ and $$e$$ is the exceptional divisor class of the blowup $$p \colon X \to \mathbb{P}^1 \times \mathbb{P}^1$$, then $$2h_2 - e$$ is ample on fibers of $$p_1$$ and with respect to it both general and special fibers have degree 2.
Finally, $$X$$ is NOT a Hirzebruch surface, since its Picard number is 3. It is the del Pezzo surface of (anticanonical) degree 7.
• Dear Sasha, thanks for your correction and explanation. Indeed, the Hirzebruch surface is blowing down a $(-1)$ curve on $X$. And I can now see that the map $\pi:X\to \mathbb P^1$ gives a family of conics degenerating to two lines. But here is a question in your answer: $2h_2-e$ is divisor on $X$ whose restriction on each fiber of $\pi$ is ample, but itself is not ample on $X$ (as its paring with $h_2$ is zero). It seems to be a notion "being ample with respect to a morphism" if I interpret correctly. It is new to me. Do you have a reference to this notion? – Yilong Zhang Jan 20 at 20:08