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The fact that the divisor "$X\,$" is not ample can also be seen directly from the definition, and does not depend on the characteristic. As usual we may assume $k$ is algebraically closed, so that $Y …

[expanding on my comments above] $H_{g,d}$ is rational for some but not all $(g,d)$. A rational example is $H_{1,4}$: a quartic elliptic curve in ${\bf P}^3$ is the complete intersection of two quad …

Well ${\bf C}[x,y]$ is isomorphic with $({\bf C}[x]) \otimes_{\bf C} ({\bf C}[y])$. But the tensor products over ${\bf R}$ cannot coincide, because already ${\bf C} \otimes_{\bf R} {\bf C}$ has zero …

Even over the complex numbers, "most" polynomials in $k$ variables are irreducible once $k \geq 2$. You can see this by comparing dimensions. The space of polynomials $P$ of degree $n$ has dimension …

The construction in that bachelor's thesis is taken directly from the 1974 paper by Shioda cited as [Sh] in the bibliography (Math. Ann. 211, 233-236), where it is shown more generally that for any pr …

There is no such curve. One way to see this is via the action of ${\rm Gal}\bigl(\overline{{\mathbb C}(t)}\big/{\mathbb C}(t)\bigr)$ on the group $E[p]$ of $p$-torsion points of a putative elliptic c …

As noted in the comments, this is a classical object, credited to [Valentiner 1889]; the paper by Robert Crass that you cite refers to that paper and also to work of Wiman [1895] and Fricke [1926] tha …

When $X$ is a curve, the Riemann-hurwitz formula is the only condition on $D$. Once $X$ has positive genus, $\deg D$ can be any even number, including zero. For example, if $X$ is an elliptic curve t …

Yes, if the quartic $C : F_4=0$ is smooth then the line $l : f_1=0$ must be one of the $28$ bitangents (because the restriction $F_4|_l$ is the square of $f_2|_l$), and then $f_2$ restricted to $l$ is …

This is essentially the question of whether a $k$-unirational variety is necessarily $k$-rational. The short answer is No. The following longer answer mostly summarizes some of the exposition at http …

Not in general. The involutions of the Markov surface such as $(a,b,c) \leftrightarrow (bc-a,b,c)$ preserve integral points because $x^2 + y^2 + z^2 - xyz$ is a monic quadratic polynomial in each vari …

The Del Pezzo connection gives another way to see that the map must be conjugate to its square and to construct an explicit conjugation. Blowing the plane up at four points in general position (i.e. …

Let $P_1,\ldots,P_8$ be "random" real points (which could even be rational). Then the space of cubic polynomials vanishing on all $P_i$ has dimension $10 - 8 = 2$. Let $(C_1, C_2)$ be a basis of this …

Yes it is possible, already for $(K,g) = ({\bf Q},2)$, and already with the first example of isogenous $C_1,C_2$ listed in the LMFDB: curve 249.a.249.1, $y^2 + (x^3+1) y = x^2 + x$, has one rational W …

Faltings' second proof extends to subvarieties of abelian varieties $A$. (The exceptional locus consists of the translates of abelian subvarieties of $A$.) That's a very special case, but it includes …