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Let me expand on my comment. Let $E$ be an elliptic curve and let $p$ be a point of $E$ of infinite order. Embed $E$ in $\mathbb{P}^2$ as a plane cubic using the linear system $3O$, where $O$ is the …

I suspect that even if you had a single curve over $\mathbb{F}_p$, you might not find a lift to $\mathbb{Q}$. Below I sketch an argument that works under the assumption that $\mathcal{M}_g$ does not …

I do not know how much progress has been made on this, but what you ask is part of a conjecture appearing in a paper of Eisenbud, Green and Harris (see Cayley-Bacharach Theorems and Conjectures, Conje …

I see that Sándor already gave an answer to the question, but I wanted to expand on my comment, giving an answer that does not rely on the Basepoint-free Theorem. It exploits the fact that ampleness …

Certainly not: even in the case of $X=Y=S=\mathbb{P}^1$ and the two maps $X,Y \to S$ are the same and general of degree at least three. In this case one component is the "diagonal" $\mathbb{P}^1$ and …

I am interpreting the question as also implying that $X$ itself is not a product. I do not have an answer in general, but I think that I have an example. It is a projective surface with a singular p …

This is mostly a series of comments, but guided by the questions you asked. First of all, I will only talk about $X_n$, interpreting it as the space of non-zero complex polynomials $p$ of degree at m …

Let $\ell_1,\ell_2,\ell_3$ be three lines in the plane that do not all contain the same point. The triangle formed by $\ell_1,\ell_2,\ell_3$ is the set obtained from $\ell_1 \cup \ell_2 \cup \ell_3$ …

This is a substantial revision of my previous answer based on the comments and the other answers. I am making it community wiki for two reasons: it incorporates ideas from my answer (for which I alrea …

A few opposite-looking remarks, and the case of (minimal) surfaces. If a variety $X$ admits a non-constant morphism to a curve, then it admits a non-constant morphism to $\mathbb{P}^1$ (just compose …

An immediate comment is that, in the case of smooth manifolds, the notion that you suggest needs to take into account more than just the differentiable structure of $X$ and $Y$: any two smooth, connec …

Here is one easy case of a curve of genus two whose Jacobian is isogenous to a product of two elliptic curves. Let $p,q,r$ be homogeneous separable polynomials of degree two in two variables $s,t$ th …

A result of Kawamata (Kawamata, Yujiro, Characterization of abelian varieties. Compositio Mathematica, 43 no. 2 (1981), p. 253-276) implies that, under your assumptions, $X$ is birational to an abelia …

A great difference in the transition from varieties to schemes is the presence of non-reducedness. Sometimes on a given scheme there are different natural scheme structures and with respect to one, t …

To complement Hailong's answer, here is another way of seeing that an irreducible component of a Cohen-Macaulay scheme need not have special properties. In the example below, the whole scheme is Cohen …