121 Patchworked Curves of Degree Seven

TL;DR

This paper presents an explicit construction method for realizing all real schemes of degree seven smooth real plane algebraic curves as T-curves, confirming a longstanding question in algebraic geometry.

Contribution

It provides a systematic patchwork construction approach for each of the 121 real schemes of degree seven curves, demonstrating their realizability as T-curves.

Findings

All 121 real schemes of degree seven are realizable as T-curves.

Constructed explicit polynomials for each real scheme.

Confirmed a question posed by Itenberg and Viro (1996).

Abstract

The 121 real schemes, i.e., ambient isotopy classes, of smooth real plane algebraic curves of degree seven were classified by Viro (1984). By constructing one patchwork of the dilated triangle $7\cdot\Delta_2$ for each real scheme, we provide an explicit method for constructing polynomials realizing each real scheme. In particular, every real scheme of degree seven can be realized as a T-curve; this settles a question raised by Itenberg and Viro (1996).