## 4. Sections and section rings

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

#### Problem 4.1.

[Cascini] Is every Fano variety a Mori Dream Space? -
### Invariance of plurigenera

#### Problem 4.2.

[Cascini] What is the status of invariance of plurigenera? -
### 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 $X$? -
### 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.
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