Contraction theorem for generalized pairs
We use Kollár's gluing theory to prove the contraction theorem for generalized pairs. In particular, we show that we can run the MMP for any generalized log canonical pairs.
math.AG↗
arXiv subjects
Lingyao Xie
Publications and source records attributed to Lingyao Xie.
On the termination of the MMP for semi-stable fourfolds in mixed characteristic
We improve on the result of Hacon and Witaszek by showing that the MMP for semi-stable fourfolds in mixed characteristic terminates in several new situations. In particular, we show the validity of the MMP for strictly semi-stable fourfolds over excellent Dedekind schemes globally when the residue fields are perfect and have characteristics $p>5$.
math.AG↗
On generalized lc pairs with $\mathrm{\textbf b}$-log abundant nef part
We study the behavior of generalized lc pairs with $\mathrm{\textbf b}$-log abundant nef part, a meticulously designed structure on algebraic varieties. We show that this structure is preserved under the canonical bundle formula and sub-adjunction formulas, and is also compatible with the non-vanishing conjecture and the abundance conjecture in the classical minimal model program.
math.AG↗
On the fixed part of pluricanonical systems for surfaces
We show that $|mK_X|$ defines a birational map and has no fixed part for some bounded positive integer $m$ for any $\frac{1}{2}$-lc surface $X$ such that $K_X$ is big and nef. For every positive integer $n\geq 3$, we construct a sequence of projective surfaces $X_{n,i}$, such that $K_{X_{n,i}}$ is ample, ${\rm{mld}}(X_{n,i})>\frac{1}{n}$ for every $i$, $\lim_{i\rightarrow+\infty}{\rm{mld}}(X_{n,i})=\frac{1}{n}$, and for any positive integer $m$, there exists $i$ such that $|mK_{X_{n,i}}|$ has non-zero fixed part. These results answer the surface case of a question of Xu.
math.AG↗
On the existence of flips for threefolds in mixed characteristic $(0,5)$
We provide a detailed proof of the validity of the Minimal Model Program for threefolds over excellent Dedekind separated schemes whose residue fields do not have characteristic 2 or 3.
math.AG↗
Rational nef and anti-nef polytopes are not uniform
We give two examples which show that rational nef and anti-nef polytopes are not uniform even for klt surface pairs, answering a question of Chen-Han. We also show that rational nef polytopes are uniform when the Cartier indices are uniformly bounded.
math.AG↗
Number of singular points on projective surfaces
The number of singular points on a klt Fano surface $X$ is $\leq 2ρ(X)+2$.
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Divisors computing minimal log discrepancies on lc surfaces
Let $(X\ni x,B)$ be an lc surface germ. If $X\ni x$ is klt, we show that there exists a divisor computing the minimal log discrepancy of $(X\ni x,B)$ that is a Kollár component of $X\ni x$. If $B\not=0$ or $X\ni x$ is not Du Val, we show that any divisor computing the minimal log discrepancy of $(X\ni x,B)$ is a potential lc place of $X\ni x$.
math.AG↗