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In this paper, we study the property of bigness of the tangent bundle of a smooth projective rational surface with nef anticanonical divisor. We first show that the tangent bundle $T_S$ of $S$ is not big if $S$ is a rational elliptic surface. We then study the property of bigness of the tangent bundle $T_S$ of a weak del Pezzo surface $S$. When the degree of $S$ is $4$, we completely determine the bigness of the tangent bundle through the configuration of $(-2)$-curves. When the degree $d$ of $S$ is less than or equal to $3$, we get a partial answer. In particular, we show that $T_S$ is not big when the number of $(-2)$-curves is less than or equal to $7-d$, and $T_S$ is big when $d=3$ and $S$ has the maximum number of $(-2)$-curves. The main ingredient of the proof is to produce irreducible effective divisors on $\mathbb{P}(T_S)$, using Serrano's work on the relative tangent bundle when $S$ has a fibration, or the total dual VMRT associated to a conic fibration on $S$., Comment: 20 pages, 14 figures

Let $\mathbf{R}_d$ be the space of stable sheaves $F$ which satisfy the Hilbert polynomial $\chi(F(m))=dm+1$ and are supported on rational curves in the projective plane $\mathbb{P}^2$. Then $\mathbf{R}_1$ (resp. $\mathbf{R}_2$) is isomorphic to $\mathbf{R}_1\cong\mathbb{P}^2$ (resp. $\mathbf{R}_2\cong \mathbb{P}^5$). Also it is very well-known that $\mathbf{R}_3$ is isomorphic to a $\mathbb{P}^6$-bundle over $\mathbb{P}^2$. In special $\mathbf{R}_d$ is smooth for $d\leq 3$. But for $d\geq4$ case, one can imagine that the space $\mathbf{R}_d$ is no more smooth because of the complexity of boundary curves. In this paper, we obtain an $\mathrm{SL}(3,\mathbb{C})$-equivariant smooth resolution of $\mathbf{R}_4$ for $d=4$, which is a $\mathbb{P}^5$-bundle over the blow-up of a Kronecker modules space., Comment: Comments are welcome

In this short article, we determine the bigness of the tangent bundle $T_X$ of the projective bundle $X=\mathbb{P}_C(E)$ associated to a vector bundle $E$ on a smooth projective curve $C$., Comment: 9 pages, error fixed

In this paper, we study the positivity property of the tangent bundle $T_X$ of a Fano threefold $X$ with Picard number 2. We determine the bigness of the tangent bundle of the whole 36 deformation types. Our result shows that $T_X$ is big if and only if $(-K_X)^3\ge 34$. As a corollary, we prove that the tangent bundle is not big when $X$ has a standard conic bundle structure with non-empty discriminant. Our main methods are to produce irreducible effective divisors on $\mathbb{P}(T_X)$ constructed from the total dual VMRT associated to a family of rational curves. Additionally, we present some criteria to determine the bigness of $T_X$., Comment: 29 pages, 2 figures. Final version to appear in Nagoya Mathematical Journal

This paper treats the strict semi-stability of the symmetric powers $S^k E$ of a stable vector bundle $E$ of rank $2$ with even degree on a smooth projective curve $C$ of genus $g \geq 2$. The strict semi-stability of $S^2 E$ is equivalent to the orthogonality of $E$ or the existence of a bisection on the ruled surface $\mathbb{P}_C(E)$ whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of $E$ with strictly semi-stable $S^3 E$. Moreover, it is shown that when $S^2 E$ is stable, every symmetric power $S^k E$ is stable for all but a finite number of $E$ in the moduli of stable vector bundles of rank $2$ with fixed determinant of even degree on $C$., Comment: 27 pages, 1 figure