Euisung Park

In this paper, we study the minimal free resolution of a nondegenerate projective variety when X is contained in a variety Y of minimal degree as a divisor. Such a variety is of interest because of its extremal behavior with respect to various properties. The graded Betti diagram of X has been completely known only when X is arithmetically Cohen‐Macaulay. Our main result in the present paper provides a detailed description of the graded Betti diagram of X for the case where X is not arithmetical

In this paper, we study how simple linear projections of some projective varieties behave when the projection center runs through the ambient space. More precisely, let $X \subset ¶^r$ be a projective variety satisfying Green-Lazarsfeld's property $N_p$ for some $p \geq 2$, $q \in ¶^r$ a closed point outside of $X$, and $X_q := π_q (X) \subset ¶^{r-1}$ the projected image of $X$ from $q$. First, it is shown that the secant locus $Σ_q (X)$ of $X$ with respect to $q$, i.e. the set of all points on

For a nondegenerate finite subscheme |$\Gamma$| in |${\mathbb P}^c$|⁠, let |${\rm reg}(\Gamma)$| and |$\ell (\Gamma)$| be, respectively, the regularity of |$\Gamma$| and the largest integer |$\ell$| such that there exists an |$\ell$|-secant line to |$\Gamma$|⁠. It is always true that |${\rm reg}(\Gamma) \geq \ell (\Gamma)$|⁠. In this article, we show that if |${\rm reg}(\Gamma) \geq \frac{d-c+5}{2}$| then |${\rm reg}(\Gamma)$| is equal to |$\ell (\Gamma)$|⁠. In addition, we describe the minimal

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X subset-of double-struck upper P Superscript r"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mi>r</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">X \subset \mathbb {P}^r</mml:annotation> </mml:semantics> </mml:math>

Abstract Classical Castelnuovo's lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension c is at most <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfenced separators="" open="(" close=")"> <m:mfrac linethickness="0pt"> <m:mrow> <m:mi>c</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mn>2</m:mn> </m:mfrac> </m:mfenced> </m:math> ${{\binom{c+1}{2}}}$ and the equality is attained if and only if the variet

For a vector bundle $\mathcal{E}$ of rank $n+1$ over a smooth projective curve $C$ of genus $g$, let $X=¶_C (\mathcal{E})$ with projection map $π:X\to C$. In this paper we investigate the minimal free resolution of homogeneous coordinate rings of $X$. We first clarify the relations between higher syzygies of very ample line bundles on $X$ and higher syzygies of Veronese embedding of fibres of $π$ by the same line bundle. More precisely, letting $H = \mathcal{O}_{¶_C (\mathcal{E})} (1)$ be the ta

Let $X \subset ¶^r$ be a nondegenerate projective variety and let $ν_{\ell} : ¶^r \to ¶^N$ be the $\ell$-th Veronese embedding. In this paper we study the higher normality, defining equations and syzygies among them for the projective embedding $ν_{\ell} (X) \subset ¶^N$. We obtain that for a very ample line bundle $L \in {Pic}X$ such that $X \subset ¶H^0 (X,L)$ is $m$-regular in the sense of Castelnuovo-Mumford, $(X,L^{\ell})$ satisfies property $N_{\ell}$ for all $\ell \geq m$ (Theorem \ref{th