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Let $X$ be a reduced scheme, so, generically regular; you may assume extra conditions like equidimensional and seminormal (though normal is stronger than I'd like, as is Gorenstein).

Is there a reasonable definition of "section of the anticanonical bundle" over $X$? At the very least I'd like to be able to restrict such a "section" to the regular locus and get an actual section. If $X$ is normal, then I'd like any (honest) anticanonical section over $X_{reg}$ to uniquely extend to one of these objects on $X$.

Note: a divisor and a section are not quite the same thing, in that the divisor of zeroes only determines the section up to a global invertible function (which need not even be constant). I really want a section, not just its divisor of zeroes.

The motivation comes from Frobenius splitting. If $X$ is defined over a perfect field of characteristic $p$, and $\sigma$ is a section of the anticanonical over $X_{reg}$ (not vanishing on any component), then away from $\sigma=0$ and the singularities we can define a morphism of sheaves $\varphi:\ F_* \mathcal O_X \to \mathcal O_X$ by $g \mapsto \mathcal C(g/\sigma) \sigma$ (here $F$ is the Frobenius and $\mathcal C$ is the Cartier operator on (top) forms). This extends over $X_{reg}$, and over all $X$ if $X$ is normal. Such maps $\varphi$ make sense even when $X$ is more singular, but I'd rather talk about an anticanonical section than about $\varphi$.

Feel free to add tags.

Let $X$ be a reduced scheme, so, generically regular; you may assume extra conditions like equidimensional and seminormal (though normal is stronger than I'd like, as is Gorenstein).

Is there a reasonable definition of "section of the anticanonical bundle" over $X$? At the very least I'd like to be able to restrict such a "section" to the regular locus and get an actual section. If $X$ is normal, then I'd like any (honest) anticanonical section over $X_{reg}$ to uniquely extend to one of these objects on $X$.

Note: a divisor and a section are not quite the same thing, in that the divisor of zeroes only determines the section up to a global invertible function (which need not even be constant). I really want a section, not just its divisor of zeroes.

The motivation comes from Frobenius splitting. If $X$ is defined over a perfect field of characteristic $p$, and $\sigma$ is a section of the anticanonical over $X_{reg}$ (not vanishing on any component), then away from $\sigma=0$ and the singularities we can define a morphism of sheaves $\varphi:\ F_* \mathcal O_X \to \mathcal O_X$ by $g \mapsto \mathcal C(g/\sigma) \sigma$ (here $F$ is the Frobenius and $\mathcal C$ is the Cartier operator on (top) forms). This extends over $X_{reg}$, and over all $X$ if $X$ is normal. Such maps $\varphi$ make sense even when $X$ is more singular, but I'd rather talk about an anticanonical section than about $\varphi$.

Feel free to add tags.

EDIT: since this hasn't produced an answer "of course you want this standard object well-known to those who know it well", here's more detail about exactly what I want.

Over $X$, we have the sheaf $\mathcal{Hom}(F_* \mathcal O_X, \mathcal O_X)$. If we restrict to $X_{reg}$, where $F$ becomes a (finite) flat morphism, then using Serre duality for such morphisms we can identify this sheaf with $F_* \mathcal{Hom}(\omega^p,\omega) = F_* \omega^{1-p}$ (as in $\S$1.3 of the book [Brion-Kumar]). Now, I'd rather work with $\omega^{-1}$, and take the $p-1$ power (which gets me only very special sections, but that's okay). But I'd like to do that on $X$, where $F$ isn't flat.

Let $X$ be a reduced scheme, so, generically regular; you may assume extra conditions like equidimensional and seminormal (though normal is stronger than I'd like, as is Gorenstein).

Is there a reasonable definition of "section of the anticanonical bundle" over $X$? At the very least I'd like to be able to restrict such a "section" to the regular locus and get an actual section. If $X$ is normal, then I'd like any (honest) anticanonical section over $X_{reg}$ to uniquely extend to one of these objects on $X$.

Note: a divisor and a section are not quite the same thing, in that the divisor of zeroes only determines the section up to a global invertible function (which need not even be constant). I really want a section, not just its divisor of zeroes.

The motivation comes from Frobenius splitting. If $X = X_{reg}$ is defined over a perfect field of characteristic $p$, and $\sigma$ is a section of the anticanonical over $X$ (not vanishing on any component), then we can define a map $\varphi:\ F_* \mathcal O_X \to \mathcal O_X$ by $g \mapsto \mathcal C(g/\sigma) \sigma$ away from $\sigma=0$, which extends over all of $X$ (here $F$ is the Frobenius and $\mathcal C$ is the Cartier operator on (top) forms). Such maps make sense even when $X$ is singular, but I'd rather talk about an anticanonical section than about $\varphi$.

If $X$ is only normal not regular, then $\mathcal O_X = \mathcal O_{X_{reg}}$, which is why I don't want to get anything new in this

Feel free to add tags.

Let $X$ be a reduced scheme, so, generically regular; you may assume extra conditions like equidimensional and seminormal (though normal is stronger than I'd like, as is Gorenstein).

Is there a reasonable definition of "section of the anticanonical bundle" over $X$? At the very least I'd like to be able to restrict such a "section" to the regular locus and get an actual section. If $X$ is normal, then I'd like any (honest) anticanonical section over $X_{reg}$ to uniquely extend to one of these objects on $X$.

Note: a divisor and a section are not quite the same thing, in that the divisor of zeroes only determines the section up to a global invertible function (which need not even be constant). I really want a section, not just its divisor of zeroes.

The motivation comes from Frobenius splitting. If $X$ is defined over a perfect field of characteristic $p$, and $\sigma$ is a section of the anticanonical over $X_{reg}$ (not vanishing on any component), then away from $\sigma=0$ and the singularities we can define a morphism of sheaves $\varphi:\ F_* \mathcal O_X \to \mathcal O_X$ by $g \mapsto \mathcal C(g/\sigma) \sigma$ (here $F$ is the Frobenius and $\mathcal C$ is the Cartier operator on (top) forms). This extends over $X_{reg}$, and over all $X$ if $X$ is normal. Such maps $\varphi$ make sense even when $X$ is more singular, but I'd rather talk about an anticanonical section than about $\varphi$.

Feel free to add tags.

Let $X$ be a reduced scheme, so, generically regular; you may assume extra conditions like equidimensional and seminormal (though normal is stronger than I'd like, as is Gorenstein).

Is there a reasonable definition of "section of the anticanonical bundle" over $X$? At the very least I'd like to be able to restrict such a "section" to the regular locus and get an actual section. If $X$ is normal, then I'd like any (honest) anticanonical section over $X_{reg}$ to uniquely extend to one of these objects on $X$.

Note: a divisor and a section are not quite the same thing, in that the divisor of zeroes only determines the section up to a global invertible function (which need not even be constant). I really want a section, not just its divisor of zeroes.

The motivation comes from Frobenius splitting. If $X = X_{reg}$ is defined over a perfect field of characteristic $p$, and $\sigma$ is a section of the anticanonical over $X$ (not vanishing on any component), then we can define a map $\varphi:\ F_* \mathcal O_X \to \mathcal O_X$ by $g \mapsto \mathcal C(g/\sigma) \sigma$ away from $\sigma=0$, which extends over all of $X$ (here $F$ is the Frobenius and $\mathcal C$ is the Cartier operator on (top) forms). Such maps make sense even when $X$ is singular, but I'd rather talk about an anticanonical section than about $\varphi$.

If $X$ is Gorenstein, then I suppose an "anticanonical section" should be a section of the dual of the canonical line bundle, but my $X$s usually aren't Gorenstein.

Feel free to add tags.

Let $X$ be a reduced scheme, so, generically regular; you may assume extra conditions like equidimensional and seminormal (though normal is stronger than I'd like, as is Gorenstein).

Is there a reasonable definition of "section of the anticanonical bundle" over $X$? At the very least I'd like to be able to restrict such a "section" to the regular locus and get an actual section. If $X$ is normal, then I'd like any (honest) anticanonical section over $X_{reg}$ to uniquely extend to one of these objects on $X$.

Note: a divisor and a section are not quite the same thing, in that the divisor of zeroes only determines the section up to a global invertible function (which need not even be constant). I really want a section, not just its divisor of zeroes.

The motivation comes from Frobenius splitting. If $X = X_{reg}$ is defined over a perfect field of characteristic $p$, and $\sigma$ is a section of the anticanonical over $X$ (not vanishing on any component), then we can define a map $\varphi:\ F_* \mathcal O_X \to \mathcal O_X$ by $g \mapsto \mathcal C(g/\sigma) \sigma$ away from $\sigma=0$, which extends over all of $X$ (here $F$ is the Frobenius and $\mathcal C$ is the Cartier operator on (top) forms). Such maps make sense even when $X$ is singular, but I'd rather talk about an anticanonical section than about $\varphi$.

If $X$ is only normal not regular, then $\mathcal O_X = \mathcal O_{X_{reg}}$, which is why I don't want to get anything new in this

Feel free to add tags.