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Just a note on the construction of the example Francesco mentioned, since in my opinion it is very instructive. Let $X$ be either the Veronese surface or a rational quartic scroll in $\mathbb P^5$. I …

The first order deformations are in general (say $Y$ has no embedded points in $f(X)$ and the image of every irreducible components of $X$ intersects the smooth locus of $Y$) parametrized by $\mathrm …

Yes. This is equivalent to the existence of a non-trivial divisor $D$ on $X$ such that there exists two distinct members of the linear system $|D|\ni D_1,D_2$ that are disjoint: $D_1\cap D_2=\emptyset …

In general, for any non-constant morphism $f:C \to \mathbb P^n$, from a $1$-dimensional Cohen-Macaulay (for instance reduced) curve $C$, one has that $$H^1(C,f^*T_{\mathbb P^n}\otimes \omega_C)=0.$$ …

This is true in a much more general setting: Fact Let $X$ be a proper variety over a noetherian ring $A$ and let $\mathscr L$ be a line bundle on $X$. Then $\mathscr L$ is ample if and only if $\ …

I suppose your assumption on $\pi$ means that $\mathcal X\subseteq \mathbb P^n\times B=:Z$ and $\pi$ is the restriction of the projection to the second factor. Then $H_B:=H\times B$ is a $\pi$-ample C …

The second short exact sequence is wrong. You should recognize this without knowing where the mistake is: $D_{\mathrm{red}}\leq D$, so $\mathscr{O}_X(D_{\mathrm{red}}) \subseteq \mathscr{O}_X(D)$ and …

Let $B={\rm Spec}\\, k[x,y]/(xy)$, i.e., the union of two lines. There is an obvious flat morphism to the line $p:B\to A={\rm Spec}\\, k[x]$. Now let $X$ be a reduced scheme. $Z=X\times B$, and $f:Z\t …

As @nfdc23 points out, even in the simplest case of $\pi=\mathrm{id}_X$ your suggestion would amount to saying that for any two locally free sheaves of the same rank (at every point) there would be a …

I think there is some confusion here. Either on your part or on mine. I don't think being non-reduced is equivalent to having a non-reduced component. A scheme may have a fat point, but be irreducible …

Angelo has already mentioned the Hilbert-Burch theorem in a comment. One could present its importance this way: The ideal of a codimension $1$ subscheme in a regular affine scheme is locally free of …

Let $S$ be the "usual" pinch point surface defined by $x^2t=y^2z$ in $\mathbb P^3$ and $T\subseteq \mathbb P^3$ an arbitrary general surface of degree $d-3\geq 2$. Let $X_0=S\cup T$. Note that then $\ …

You already have trivial counterexamples for your statement, but perhaps you were thinking of a section whose zero locus is irreducible and dominates $Y$. It is false even with that additional assumpt …