Characterization of projective varieties beyond varieties of minimal degree and del Pezzo varieties

TL;DR

This paper extends the classification of basic projective varieties by establishing bounds on Betti numbers, characterizing extremal cases, and linking these varieties to rational normal scrolls, broadening understanding beyond minimal degree and del Pezzo varieties.

Contribution

It provides new bounds on graded Betti numbers, characterizes extremal varieties beyond classical types, and relates these varieties to rational normal scrolls, generalizing previous classifications.

Findings

Characterized extremal varieties with specific degree and codimension.

Bounded graded Betti numbers in the quadratic strand.

Connected extremal varieties to rational normal scrolls.

Abstract

Varieties of minimal degree and del Pezzo varieties are basic objects in projective algebraic geometry. Those varieties have been characterized and classified for a long time in many aspects. Motivated by the question "which varieties are the most basic and simplest except the above two kinds of varieties in view of geometry and syzygies?", we give an upper bound of the graded Betti numbers in the quadratic strand and characterize the extremal cases. The extremal varieties of dimension $n$, codimension $e$, and degree $d$ are exactly characterized by the following two types: (i) varieties with $d = e+2$, $\operatorname{depth} X =n$, and Green-Lazarsfeld index $a(X)=0$, (ii) arithmetically Cohen-Macaulay varieties with $d = e+3$. This is a generalization of G. Castelnuovo, G. Fano, and E. Park's results on the number of quadrics and an extension of the characterizations of varieties of minimal degree and del Pezzo varieties in view of linear syzygies of quadrics due to K. Han and S. Kwak. In addition, we show that every variety $X$ that belongs to (i) or (ii) is always contained in a unique rational normal scroll $Y$ as a divisor. Also, we describe the divisor class of $X$ in $Y$.