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4. Sections and section rings

Questions about the existence of sections, and properties of various section rings.
1. Fano and MDS

Problem 4.1.

[Cascini] Is every Fano variety a Mori Dream Space?
This result is known in characteristic $0$. It’s also true that if $X$ is Fano then $X$ is strongly $F$-regular, if $p \gg 0$.
• Invariance of plurigenera

Problem 4.2.

[Cascini] What is the status of invariance of plurigenera?
It’s known that $h^0(mK_X)$ need not be constant in flat families if $m=1$. For $m \geq 2$ it remains a possibility.
• Problem 4.3.

[Hacon] What if we look at $\text{dim } S^0(mK_X+A)$ instead?
• General type in families

Problem 4.4.

[McKernan] Suppose that $\mathcal X \to \Delta$ is a family, and a special fiber is of general type. Does it follow that a general fiber is of general type?
• Embedding by $S^0(mK_X)$

Problem 4.5.

[McKernan] Suppose that $X$ is of general type. Is there an effective constant $m = m(n)$ such that $S^0(mK_X)$ (the canonical linear system of Schwede) defines a birational map? What about the usual linear system $|mK_X|$?
• Effective Fujita vanishing

Problem 4.6.

[McKernan] Is there an effective Fujita vanishing result in flat families? Suppose that $f : \mathcal X \to S$ is flat, $\mathcal F$ is a coherent sheaf on $\mathcal X$, and $\mathcal L$ is $f$-ample. Does there exist a constant $m_0 = m_0(\mathcal X,\mathcal F,\mathcal L)$ such that $H^i(X_s,(\mathcal F \otimes \mathcal L^m \mathcal )\vert_{X_s} ) = 0$ if $s$ is any point of $S$, $m \geq m_0$ and $\mathcal M$ is any $f$-nef line bundle on $X$?
This is probably OK in the equal characteristic case, adapting the usual proof.
• Sections of nef $K_X+A$

Problem 4.7.

[Cascini] Suppose that $A$ is Cartier and ample, and $K_X+A$ is nef. Must $H^0(X,K_X+A)$ be nonzero?
• $F$-regularity and finite generation

Problem 4.8.

[Schwede] Suppose that $X$ is an $F$-regular variety (maybe not $\mathbb Q$-Gorenstein). If $D$ is a Weil divisor, must $\bigoplus_{m \geq 0} \mathcal O_X(mD)$ be finitely generated?

Cite this as: AimPL: The minimal model program in characteristic p, available at http://aimpl.org/minimalmodcharp.