arXiv subjects
Publications and source records attributed to Jiayuan Lin.
It is well known that Thom-Boardman symbols are realized by non-increasing sequences of nonnegative integers. A natural question is whether the converse is also true. In this paper we answer this question affirmatively, that is, for any non-increasing sequence of nonnegative integers, there is a map-germ with the prescribed sequence as its Thom-Boardman symbol.
In [7] we obtained a formula for the Hilbert depth of squarefree Veronese ideals in a standard graded polynomial ring by relating it to the Hilbert depth of powers of the irrelevant maximal ideal. In this paper, we prove that these two Hilbert depth formulas are equivalent to each other. Our result reveals that there is a strong connection between these two classes of seemingly unrelated monomial ideals. We conjecture that their Stanley depths are equivalent as well.
We provide a natural smooth projective compactification of the space of algebraic maps from the projective line to the projective space of dimension n by adding a divisor with simple normal crossings.
In this paper, we obtain explicit formulas for the Hilbert series and Hilbert depth of squarefree Veronese ideals in a standard graded polynomial ring.
In this paper, we partially confirm a conjecture, proposed by Cimpoeaş, Keller, Shen, Streib and Young, on the Stanley depth of squarefree Veronese ideals $I_{n,d}$. This conjecture suggests that, for positive integers $1 \le d \le n$, $\sdepth (I_{n,d})= \lfloor \binom{n}{d+1}/\binom{n}{d} \rfloor+d$. Herzog, Vladoiu and Zheng established a connection between the Stanley depths of quotients of monomial ideals and interval partitions of certain associated posets. Based on this connection, Keller, Shen, Streib and Young recently developed a useful combinatorial tool to analyze the interval partitions of the posets associated with the squarefree Veronese ideals. We modify their ideas and prove that if $1 \le d \le n \le (d+1) \lfloor \frac{1+\sqrt{5+4d}}{2}\rfloor+2d$, then $\sdepth (I_{n,d})= \lfloor \binom{n}{d+1}/\binom{n}{d} \rfloor+d$. We also obtain $ \lfloor \frac{d+\sqrt{d^2+4(n+1)}}{2} \rfloor \le \sdepth(I_{n,d}) \le \lfloor \binom{n}{d+1}/\binom{n}{d} \rfloor+d$ for $n > (d+1) \lfloor \frac{1+\sqrt{5+4d}}{2}\rfloor+2d$. As a byproduct of our construction, We give an alternative proof of Theorem $1.1 $ in $[13]$ without graph theory.
The Thom-Boardman symbol was first introduced by Thom in 1956 to classify singularities of differentiable maps. It was later generalized by Boardman to a more general setting. Although the Thom-Boardman symbol is realized by a sequence of non-increasing, nonnegative integers, to compute those numbers is, in general, extremely difficult. In the case of polynomial multiplication maps, Robert Varley conjectured that computing the Thom-Boardman symbol for polynomial multiplication reduces to computing the successive quotients and remainders for the Euclidean algorithm applied to the degrees of the two polynomials. In this paper, we confirm this conjecture.
Let $M_{d}(¶^r)$ be the space of $(r+1)$-tuples $(f_0,...,f_r)$ modulo homothety, where $f_0,...,f_r$ are homogeneous polynomials of degree $d$ in two variables. Let $M_{d}^{\circ}(¶^r)$ be the open subset of $M_{d}(¶^r)$ such that $f_0,...,f_r$ have no common zeros. Then $M_{d}^{\circ}(¶^r)$ parametrizes the space of holomorphic maps of degree $d$ from $¶^1$ into $¶^r$. In general the boundary divisor $M_{d}(¶^r) \setminus M_{d}^{\circ}(¶^r)$ is not normal crossing. In this paper we will give a natural stratification of this boundary and show that we can process an iterated blow-ups along these strata (or its proper transformations) to obtain a compactification of $M_{d}^{\circ}(¶^n)$ with normal crossing divisors.
In this paper, we prove that the family of $\Bbb Q$-Fano threefolds with Picard number one is birationally unbounded.
Let $\Hol_{x_0}^{\bf n} (\C¶^1, X)$ be the space of based holomorphic maps of degree ${\bf n}$ from $\C¶^1$ into a simply connected algebraic variety $X$. Under some condition we prove that the map $\map \Hol_{x_0}^{\bf n} (\C¶^1, X). \Hol_{x_0}^{d{\bf n}} (\C¶^1, X).$ obtained by compositing $f \in \Hol_{x_0}^{\bf n} (\C¶^1, X)$ with $g(z)=z^d, z \in \C¶^1$ induces rational homotopy equivalence up to some dimension, which tends to infinity as the degree grows.